Bridging Theory and Algorithm for Domain Adaptation
Bridging Theory and Algorithm for Domain Adaptation
Yuchen Zhang, Tianle Liu, Mingsheng Long, Michael Jordan
Preliminaries
首先是定义margin，使用margin loss，而不是0／1 loss。（其实这个Behnam generalization bound中也用过）
$$ \rhof(x,y) = \frac{1}{2} \big( f(x, y) - \max{y' \ne y} f(x,y') \big)
$$
那么margin loss和empirical margin loss就定义如下
$$ errD(f) = \mathbb{E}{x \sim D} \Phi\rho \circ \rho_f(x,y)\ err{\hat D}(f) = \mathbb{E}{x \sim \hat D} \Phi\rho \circ \rhof(x,y) = \frac{1}{n} \sum{i=1}^n \Phi_\rho ( \rho_f(x_i,y_i) )
$$
其中$$\circ$$表示的是function composition，而$$\Phi$$是
$$ \Phi_\rho(x) = \begin{cases} 0 & \rho \le x\ 1-x/\rho & 0 \le x \le \rho\ 1 & x \le 0 \end{cases}
$$