Prove logistic loss has a global minimum

NLL with sigmoid: J(theta) = -sum [y_i log sigma(theta x_i) + (1-y_i) log (1 - sigma(theta x_i))]. Equivalent: -sum [y_i theta^T x_i - log(1 + exp(theta^T x_i))]. The term inside is the log-partition; its Hessian is X^T W X with W = diag(sigma(1-sigma)) >= 0, so the Hessian is positive semi-definite everywhere, so J is convex, so any stationary point is a global min. Strict convexity requires X to be full column rank; otherwise the minimum is a manifold.