By Johan A. K. Suykens
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Extra resources for Advances in learning theory: methods, models, and applications
C Sn... 24) and S* — Ufc Sk- An admissible structure is one that satisfies the following three properties: 1. The set S* is everywhere dense in S. 2. The VC-dimension hk of each set Sk of functions is finite. 3. Any element Sk of the structure contains totally bounded functions 0 < Q(z, a) < Bk, a e A f c . 20) is minimal. The SRM principle actually suggests a trade-off between the quality of the approximation and the complexity of the approximating function. 20)) increases. The SRM principle takes both factors into account.
Let v*(m, 5) be this solution. Then, also by Lemma 7, and and we can conclude stating the following result. Theorem 3 Given m>l and 0 < S < I, for all 7 > 0, the expression bounds the sample error with confidence at least 1 — 8. 5 F. Cucker, S. Smale Choosing the optimal 7 We now focus on the approximation error £(/7). g. , V(\\f II*' P II Since the minimum above is attained at /7 we deduce A basic result in [CS] (Proposition 1, Chapter I) states that, for all / € -/p) 2 + <7p (2-4) where d^ is a non-negative quantity depending only on p.
Smale Prob PROOF. -2e r m7*— 1=1 Consider the random variable ^7 It is almost everywhere bounded by ^C#Mp. Its mean is 0 since, by Fubini's Theorem f 1 I -K(x,t}(fp(x] JzJ -y) = [ I f f \ -K(x,t) I fp(x) - y dp(y\x] I dpx JxJ \JY ) and the inner integral is 0 by definition of fp. Now apply Hoeffding's inequality. D Lemma 4 For all 7, e > 0 and all t e X, Prob PROOF. A(*) 1 meV I : € V > 1 - 2e 1=1 By Theorem 1, + 7/7 = The function inside the last integral can be thus considered a random variable on X with mean /7(t).