Try to plot squared cost function for logistic regression, trying to prove that it is non-convex

square-cost.py

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import math

#==============================================================================
# Try to verify the cost function is non-convex

xs = np.random.rand(30)
ys = np.random.rand(30)
labels = np.array([1 if x + y > 1 else 0 for x, y in zip(xs, ys)])

def sigmoid(x, y, t1, t2):
    return 1/(1 + math.e**(-t1*x -t2*y))

def cost(t1,t2):
    diffs = [(sigmoid(x,y, t1,t2) - z) for x,y,z in zip(xs,ys,labels)]
    return sum(0.5 * diff * diff for diff in diffs)/500

x = y = np.arange(-20, 20, 0.5)
X, Y = np.meshgrid(x, y)
zs = np.array([cost(x, y) for x, y in zip(np.ravel(X), np.ravel(Y))])
Z = zs.reshape(X.shape)

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X, Y, Z)
plt.show()

Here is a sample plot. May not be obvious enough, but there is a "flat" area.

Note that here is another source that give examples on how it is non-convex: https://math.stackexchange.com/questions/2381724/logistic-regression-when-can-the-cost-function-be-non-convex/2381748#2381748?newreg=5cbf2a241e8b4a18ac3b434a402ed51a

Here is 1D version

samples = [(-5, 1), (-20, 0), (-2, 1)]

def sigmoid(theta, x):
    return 1/(1 + math.e**(- theta*x))

def cost(theta):
    diffs = [(sigmoid(theta, x) - y) for x,y in samples]
    return sum(diff * diff for diff in diffs)/len(samples)/2

X = np.arange(-1, 1, 0.01)
Y = np.array([cost(theta) for theta in X])
plt.plot(X, Y)
plt.show()