Logistic Regression

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

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者y

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= 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)−y⃗ )