By Johan A. K. Suykens

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I=l 37 Best Choices for Regularization Parameters in Learning Theory Note that, since max{Mp, ||/p||oo + x/Ctf^y} < M -f 7 the confidence above is at least Applying this to £ = x\,... ,xm and writing the m resulting inequalities in matrix form we obtain that, with confidence at least the one in the statement, 1 1 rrry <2e. D Lemma 6 For all 7, e > 0, 7m / " PROOF. , xm and writing the resulting m equalities in matrix form we obtain 7"^/7,z[x] + /^ztxjATfx] = K[x]y from which the statement follows.

The bound £(7) found is a natural one, a bound which flows from the hypotheses and thus yields a 7* which could be useful in the algorithmics for /7iZ. Of course, 7* depends on the number of examples m, confidence 1 — 6, as well as the operator A and a simple invariant of p. 2 RKHS and Regularization Parameters Let X be a compact domain or a manifold in Euclidean space and Y = 1R (one can extend all what follows to Y = Hfc with k € IN). Let p be a Borel probability measure on Z = X x Y. t. e. the measure on X defined by Px(S) = p(7r~1(S)) where TT : X x Y —>• X is the projection.

4) for / = /7iZ, This proves the first part of the Main Result. Note that this is actually a family of bounds parameterized by t < 2 and 0 < 0 < I and depends on m, S, K and fp. For a point 7 > 0 to be a minimum of E(i] = ^(7) + ^(7) it is necessary that ) =0. e. those of - 64MV(rM)C*(7 + Cir) = 0 7,,+2 _ 7 ~ ~ 64MV(m,f)C%. ~ ( ] Using again Lemma 7, we obtain a unique solution 7* which is a minimize! of E since » o o a s 7 —» 0 o r 7 —>oo. This finishes the proof of the Main Result. D Corollary 2 For every 0 < o < I , lim E(f) = 0.