使hypotheses hθ(x) to satisfy 0≤hθ(x)≤1.

z > 0,g(z) > 0.5 ,y=1;

z< 0,g(z) < 0.5 ,y=0;

Cost Function:

When y = 1, we get the following plot for J(θ) vs hθ(x):

Similarly, when y = 0, we get the following plot for J(θ) vs hθ(x):

Cost(hθ(x),y) = 0 if hθ(x) = y;

Cost(hθ(x),y) ->∞ if y = 0 and hθ(x) ->1 或

者y

= 1 and hθ(x) ->0.

Simplified Cost Function:

 Cost(hθ(x),y)=?ylog(hθ(x))?(1?y)log(1?hθ(x))
y = 1 时，Cost(hθ(x),y) = ?log(hθ(x))；

y = 0时， Cost(hθ(x),y) = -log(1?hθ(x))；

Gradient Descent

向量化：θ:=θ?（α/m） XT(g(Xθ)?y? )

Logistic Regression