Gradient descent turns the p = 0.5 line into place.
Learning rate Step , loss , accuracy
Gradient descent on the log-loss turns and shifts the line where p = 0.5 until it splits the two classes as well as a straight line can. Amber points are class 1, blue class 0, red rings are on the wrong side, and the dashed lines mark p = 0.1 and 0.9. Change the learning rate and watch the same descent crawl or snap into place; the points are illustrative.