The error rate of learning halfspaces using kernel-SVM

Denote ||g|| Hk = C. By Lemma 5.25, it holds thatC ≤ 2m1.5|∂ + l(0)| Err µ N ,l(g)≤ 2m1.5 Err µ,l (g)|∂ + l(0)| λ N≤ 2m1.5 l(0)|∂ + l(0)| λ NAs in the proof of Theorem 2.6, it holds that∫∫∣ g −g∣ ≤ 128l(0)γK3.5 + 10 · K 3.5 · E · C · (r K + s d ) (19)|∂ + l(0)|λ 3{x:〈x,e〉=γ}{x:〈x,e〉=−γ}Denote the last bound by ɛ. It holds thatErr D,l (g) = (1 − λ 2 − λ 3 − λ N )E µ 1 el(yg(x)) + λ 2 E µ 2 el(yg(x)) + λ 3 E µ 3 el(yg(x)) + λ N E µN l(yg(x))(20)Now, denote δ = ∫ g. It holds that{x:〈x,e〉=−γ}∫∫E µ 1 el(yg(x)) = θ l(g(x)) + (1 − θ)l(−g(x))Thus,≥≥{x:〈x,e〉=γ}(∫θ · lg{x:〈x,e〉=γ})+ (1 − θ) · lθ · l(δ) + (1 − θ) · l(−δ) − Lɛ{x:〈x,e〉=−γ}(−∫{x:〈x,e〉=−γ}Err D,l (g) ≥ (1 − λ 2 − λ 3 − λ N )(θ · l(δ) + (1 − θ) · l(−δ)) − Lɛ + λ 2 E µ 2 el(yg(x))However, by considering the constant solution δ, it follows thatErr D,l (g) ≤ (1 − λ 2 − λ 3 − λ N )(a · l(δ) + (1 − θ) · l(−δ)) + λ 2 · l(δ) + (λ 3 + λ N ) 1 2 (l(δ) + l(−δ)) + √ γThus,≤)g(21)(1 − λ 2 − λ 3 − λ N )(θ · l(δ) + (1 − θ) · l(−δ)) + λ 2 · l(δ) + (λ 3 + λ N ) · l(−|δ|) + √ γErr µ 2 e ,l(g) ≤ Lɛλ 2+ l(δ) + λ 3 + λ Nλ 2l(−|δ|) +≤L · l(0)128γK3.5|∂ + l(0)|λ 2 λ 3+10 · L · K3.5λ 2√ γλ 2(22)· E · C · (r K + s d ) + l(δ) + λ 3 + λ N + √ γl(−|δ|)λ 2Now, relying on the assumption that γ · log 8 (C) = o(1), it is possible to choose λ 2 =Θ ( √ γK4 ) = Θ ( √ γ log 4 (C) ) , λ 3 = √ γ, K = Θ(log(C/γ)), λ N = γ and d = Θ(log(C/γ))such that the bound in Equation (19), L·l(0)128γK3.5λ 3 +λ N + √ γλ 2are all o(1).|∂ + l(0)|λ 2 λ 3+ 10K3.5λ 240· E · C · (r K + s d ), λ 2 , λ 3 , λ N and