Chapter 4
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Series Foreword / Preface / Chapter 1 / Chapter 2 / Chapter 3 / Chapter 4 / Chapter 5 / References

 

Bibliographical Remarks

This chapter reviewed different mathematical models for learning. We demonstrated that classical statistical analysis is not suited for the purpose of learning because it studies the convergence of probability measures (see Billingsley (1968), Pollard (1984) and Amari (1985)) and thus leads to observations such as the "curse of dimensionality'' (Bellman 1961). Further, classical statistical results often have to assume the "correctness'' of the probabilistic model which is essential for the maximum likelihood method to provide good convergence results (see Devroye et al. (1996, Chapters 15, 16) for a discussion with some pessimistic results). In contrast, it has been suggested that studying convergence of risks directly is preferable (see Vapnik and Chervonenkis (1971), Vapnik (1982), Kearns and Vazirani (1994), Devroye et. al (1996), Vidyasagar (1997), Anthony (1997), Vapnik (1998) and Anthony and Bartlett (1999)). In the case of empirical risk minimization algorithms this has resulted in the so-called VC and PAC framework. The PAC framework was introduced 1984 in the seminal paper of Valiant (1984) in which he specializes the general question of convergence of expected risks to the problem of learning logic formulas assuming that the hypothesis space H contains the target formula. Hence all uncertainty is due to the unknown input distribution PX. The restriction to logic formulas also simplified the matter because the number of hypotheses then becomes finite even though it grows exponentially in the number of binary features. Since then a number of generalizations have been proposed by dropping the assumption of finite hypothesis spaces and realizability, i.e., "oracle'' draws its target hypothesis h* from the hypothesis space H which we use for learning (see Blumer et al. (1989) and Anthony (1997) for a comprehensive overview). The latter generalization became known as the agnostic PAC framework Kearns et al. (1992). Though we have ignored computational complexity and computability aspects, the PAC model in its pure form is also concerned with these questions.

Apart from these developments, V. Vapnik and A. Chervonenkis already studied the general convergence question in the late 1960s. In honor of them, their framework is now known as the VC (Vapnik-Chervonenkis) framework. They showed that the convergence of expected risks is equivalent to the uniform convergence of frequencies to probabilities over a fixed set of events (Vapnik and Chervonenkis 1991) (see Vapnik (1998, Chapter 16) for a definition of "nontrivial'' hypothesis spaces and Bartlett et al. (1996) for a constructive example). This equivalence is known as the key theorem in learning theory. The answer to a particular case of this problem was already available through the Glivenko-Cantelli lemma (Glivenko 1933, Cantelli 1933) which says that the empirical distribution function of a one dimensional random variable converges uniformly to the true distribution function in probability. The rate of convergence was proven for the first time in Kolmogorov (1933). Vapnik and Chervonenkis generalized the problem and asked themselves which property a set of events must share such that this convergence still takes place. As a consequence, these sets of events are known as Glivenko-Cantelli. In 1987, M. Talagrand obtained the general answer to the problem of identifying Glivenko-Cantelli classes Talagrand (1987). Ten years later this result was independently rediscovered by Alon et al. (1997). It is worth mentioning that most of the results in the PAC framework are particular cases of more general results already obtained by Vapnik and coworkers two decades before.

The main VC and PAC bounds given in equations (4.11) and (4.12) were first proven in Vapnik and Chervonenkis (1974) and effectively differ by the exponent at the deviation of e. In Vapnik (1982, Theorem 6.8) it is shown that this exponent continously varies from 2 to 1 w.r.t. the smallest achievable expected risk infhH R[h] (see also Lee et al. (1998) for tighter results in the special case of convex hypothesis spaces). The VC and PAC analysis revealed that, for the case of learning, the growth function of a hypothesis space is an appropriate a-priori measure of its complexity. As the growth function is very difficult to compute, it is often characterized by a one-integer summary known as VC dimension (see Theorem 4.10 and Sontag (1998) for an excellent survey of the VC dimension). The first proof of this theorem is due to Vapnik and Chervonenkis (1971) and was discovered independently in Sauer (1972) and Shelah (1972); the former credits Erdös with posing it as a conjecture. In order to make the VC dimension a variable of the learning algorithm itself two conceptually different approaches were presented: By defining an a-priori structuring of the hypothesis space—sometimes also referred to as a decomposition of the hypothesis space H (Shawe-Taylor et al. 1998)—it is possible to provide guarantees for the generalization error with high confidence by sharing the confidence among the different hypothesis spaces. This principle, known as structural risk minimization, is due to Vapnik and Chervonenkis (1974). A more promising approach is to define an effective complexity via a luckiness function which encodes some prior hope about the learning problem given by the unknown PZ. This framework, also termed the luckiness framework is due to Shawe-Taylor et al. (1998). For more details on the related problem of conditional confidence intervals the interested reader is referred to Brownie and Kiefer (1977), Casella (1988), Berger (1985) and Kiefer (1977). All examples given in Section 4.3 are taken from Shawe-Taylor et al. (1998). The luckiness framework is most advantageous if we refine what is required from a learning algorithm: A learning algorithm A is given a training sample z Zm and a confidence [0,1], and is then required to return a hypothesis A(z) H together with an accuracy e such that in at least 1- of the learning trials the expected risk of A(z) is less than or equal to the given e. Y. Freund called such learning algorithms self bounding learning algorithms (Freund 1998). Although, without making explicit assumptions on PZ, all learning algorithms might be equally good, a self bounding learning algorithm is able to tell the practitioner when its implicit assumptions are met. Obviously, a self bounding learning algorithm can only be constructed having a theoretically justified generalization error bound available.

In the last section of this chapter we presented a PAC analysis for the particular hypothesis space of linear classifiers making extensive use of the margin as a data dependent complexity measure. In Theorem 4.25 we showed that the margin, that is, the minimum real-valued output of a linear classifier before thresholding, allows us to replace the coarse application of the union bound over the worst case diversity of the binary-valued function class by a union bound over the number of equivalence classes witnessed by the observed margin. The proof of this result can also be found in Shawe-Taylor and Cristianini (1998, Theorem 6.8) and Bartlett (1998, Lemma 4). Using a scale sensitive version of the VC dimension known as the fat shattering dimension (Kearns and Schapire 1994) we obtained bounds on the expected risk of a linear classifier which can be directly evaluated after learning. An important tool was Lemma 4.29 which can be found in Alon et al. (1997). The final step was an application of Lemma 4.31 which was proven in Gurvits (1997) and later simplified in Bartlett and Shawe-Taylor (1999). It should be noted, however, that the application of Alon's result yields bounds which are practically irrelevant as they require the training sample size to be of order 105 in order to be nontrivial. Reinterpreting the margin we demonstrated that this margin bound directly gives a bound on the expected risk involving a function of the margin distribution. This study closely followed the original papers Shawe-Taylor and Cristianini (1998) and Shawe-Taylor and Cristianini (2000). A further application of this idea showed that although not containing any margin complexity, adaptive margin machines effectively minimize the complexity of the resulting classification functions. Recently it has been demonstrated that a functional analytic viewpoint offers ways to get much tighter bounds on the covering number at the scale of the observed margin (see Williamson et al. (2000) Shawe-Taylor and Williamson (1999), Schölkopf et al. (1999) and Smola et al. (2000)).