By Johan A. K. Suykens
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Additional info for Advances in learning theory: methods, models, and applications
2 Equivalent definition of the VC dimension In this section we give an equivalent definition of the VC dimension of sets of indicator functions and we generalize this definition to sets of real functions. The VC dimension of a set of indicator functions. , Zh which can be separated in all 1h possible ways using functions of this set8 (shattered by this set of functions). If for any n there exists a set of n vectors which can be shattered by the set Q(z, a), a € A, then the VC-dimension is equal to infinity.
To minimize the risk in this case it is necessary to find a method which, along with minimizing the value of empirical risk, controls the VC-dimension of the learning machine. The following principle, which is called the principle of Structural Risk Minimization (SRM), is intended for minimizing the risk functional with respect to both empirical risk and VC-dimension of the set of functions. Let, the set S of functions Q(z,a), a e A, 10 The sample size t is considered to be small if t/h is small, say i/h < 20.
20) becomes small. The actual risk is then close to the value of the empirical risk. In this case, a small value of the empirical risk provides a small value of (expected) risk. However, if tfh is small, then even a small Remp(&i) does not guarantee a small value of risk. 20), one of which depends on the value of the empirical risk while the second depends on the VC-dimension of the set of functions. To minimize the risk in this case it is necessary to find a method which, along with minimizing the value of empirical risk, controls the VC-dimension of the learning machine.
Advances in learning theory: methods, models, and applications by Johan A. K. Suykens
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