The error rate of learning halfspaces using kernel-SVM

Then, for every K ∈ N, 1 8 > γ > 0,|f(γ) − f(−γ)| ≤ 32γK 3.5 · ||f|| 1,µ + ( 32γK 3.5 + 2 ) · C · E · (r K + s d )Here, E, r and s are the constants from Lemma 5.19.Proof Let ¯f = ∑ K−1n=0 α nP d,n . We have || ¯f|| 1,µ ≤ ||f|| 1,µ +|| ¯f −f|| ∞,µ . Define g : [−1, 1] → Rby g(x) = ¯f( x ) and denote by dλ =dx8 π √ . Write,1−x 2g =K−1∑n=0β n T nWhere T n are the Chebyshev polynomials. By Lemma 5.20 it holds that, for every 0 ≤ n ≤K − 1,|β n | ≤ √ 2||g|| 2,λ ≤ 2 √ K||g|| 1,λ = 2 √ K|| ¯f|| 1,µNow,g ′ =K−1∑n=1β k nU n−1Where U n are the Chebyshev polynomials of the second kind. Thus,||g ′ || ∞,λ ≤Finally, by Lemma 5.19,K−1∑n=1|β k | · n · ||U n−1 || ∞,λ =K−1∑n=1|β k | · n 2 ≤ 2 √ K|| ¯f|| 1,µ · K 3|f(γ) − f(−γ)| ≤ |g(8γ) − g(−8γ)| + 2||f − ¯f|| ∞,µ≤≤32γK 3.5 · || ¯f|| 1,µ + 2||f − ¯f|| ∞,µ32γK 3.5 · (||f||1,µ + ||f − ¯f||)∞,µ + 2||f − ¯f||∞,µ≤32γK 3.5 · ||f|| 1,µ + ( 32γK 3.5 + 2 ) · ||f − ¯f|| ∞,µ≤ 32γK 3.5 · ||f|| 1,µ + ( 32γK 3.5 + 2 ) · E · C · (r K + s d )For e ∈ S d−1 we define the group O(e) := {A ∈ O(d) : Ae = e}. If H k be a symmetricRKHS and e ∈ S d−1 we define Symmetrization around e. This is the operator P e : H k → H kwhich is the projection on the subspace {f ∈ H k : ∀A ∈ O(e), f ◦ A = f}. It is not hardto see that (P e f)(x) = ∫ {x ′ :〈x ′ ,e〉=〈x,e〉} f(x′ )dx ′ = ∫ f ◦ A(x)dA. Since P O(e) ef is a convexcombination of the functions {f ◦ A} A∈O(e) , it follows that if R : H k → R is a convexfunctional then R(P e f) ≤ ∫ R(f ◦ A)dA.O(e)Lemma 5.22 (main) There exists a probability measure µ on [−1, 1] with the followingproperties. For every continuous and normalized kernel k : S d−1 × S d−1 → R and C > 0,there exists e ∈ S d−1 such that, for every f ∈ H k with ||f|| Hk ≤ C, K ∈ N and 0 < γ < 1,8 ∫∫∣ f −f∣ ≤ 32γK 3.5 · ||f|| 1,µe + ( 32γK 3.5 + 2 ) · E · C · (r K + s d ){x:〈x,e〉=γ}{x:〈x,e〉=−γ}≤ 32γK 3.5 · ||f|| 1,µe + 10 · E · K 3.5 · C · (r K + s d )30✷