证明:
$S_{n}=sum_{i=1}^{n}Z_{i}$
Hoeffding不等式:$P(ES_{n}-S_{n}ge t)le e^{-frac{2t^{2}}{sum(b_{i}-a_{i})^{2}}} $
(其中$[b_{i},a_{i}]$是$Z_{i}$所属的范围,下面这个问题里设为[0,1])
$令Z_{i}=L(Y_{i},f(X_{i})),则S_{n}=sum_{i=1}^{n}Z_{i}=sum_{i=1}^{n}L(Y_{i},f(X_{i}))$
$E(S_{n})=nEZ=nEL(Y_{i},f(X_{i}))$带入Hoeffding不等式:
$P(nEL(Y_{i},f(X_{i}))-sum_{i=1}^{n}L(Y_{i},f(X_{i})) ge t) le e^{-frac{2t^2}{n}}$
$P(R(f)-widehat{R}(f)ge frac{t}{n})le e^{-frac{2t^2}{n}}$
$令s=frac{t}{n}带入即得。$