Logic regression of "machine learning"

What we want is to receive all the input and then predict the category. Output 0 or 1 in the case of two classes. A function of this nature, perhaps the original you have contacted, is called the Heaviside step function, that is, the unit steps functions. But this momentary jump is actually difficult to deal with. So, here we take the sigmoid function.
G (z) =11+e?z

In order to achieve the logistic regression classifier, we take a regression coefficient in each feature, add all the results, and bring this sum into the sigmoid function to get a numerical value between the 0~1. Any number greater than 0.5 is divided into 1 categories, and less than 0.5 is divided into 0 categories. So the question becomes what is the optimal regression coefficient?

To get the optimal value of the function, we need to know the direction of the optimal value and the moving quantity that needs to be moved to the optimal value. Gradient operator is the fastest way to decrease function value. The move is called the step length, which is the alpha.
With a set of training set {(x1,y1),..., (Xm,ym)},m is the sample number, for each sample xi,xi=[xi1,... xin]t,y=[0,1], we multiply the best coefficient theta to get
hθ (x) =11+e?θtx
The equation for the cost function is as follows:
Cost (hθ (x), y) ={?log (hθ (x)), log (1?hθ (x)) ify=1ify=0
J (θ) =1m∑i=0mcostfunction (hθ (xi), Yi) =?1m[∑i=0myilog (hθ (xi) + (1?yi) log (1?hθ) (xi))]