Logistic regression Prove that the cost function is convex - Mathematics Stack Exchange

Logistic regression cost function. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Logistic regression: Prove that the cost function is convex. You can do a find on "convex" to see the part that relates to my question. Background: $h_\theta(X) = sigmoid(\theta^T X)$ --- hypothesis/prediction function $y \in \ $ Normally, we would have the cost function for one sample $(X,y)$ as: $y(1 - h_\theta(X))^2 + (1-y)(h_\theta(X))^2$ It's just the squared distance from 1 or 0 depending on y. However, the lecture notes mention that this is a non-convex function so it's bad for gradient descent (our optimisation algorithm). So, we come up with one that is supposedly convex: $y * -log(h_\theta(X)) + (1 - y) * -log(1 - h_\theta(X))$ You can see why this makes sense if we plot -log(x) from 0 to 1: i.e. if y = 1 then the cost goes from $\infty$ to 0 as the hypothesis/prediction moves from 0 to 1. My question is: How do we know that this new cost function is convex? Here is an example of a hypothesis function that will lead to a non-convex cost function: $h_\theta(X) = sigmoid(1 + x^2 + x^3)$ leading to cost function (for y = 1): $-log(sigmoid(1 + x^2 + x^3))$ which is a non-convex function as we can see when we graph it: