How many minima does the residual sum of squares have for the logistic curve? Suppose we had some data points $\{(x_i,y_i)\}$ where $x_i$ is a real number and $y_i$ is zero or one. We want to fit the logistic function to the data where the logistic function is $\hat{y}=\frac{1}{1+e^{-(\beta x +\beta_0)}}$. To do this we would like to choose $\beta_1$ and $\beta_0$ in order to minimize the residual sum of squares, $RSS = \sum_i (y_i-\hat{y}_i)^2$.[nevermind that cross-entropy is a better cost function.] I believe there is only a single minimum. How can I know? Does anyone know of a proof for this or a counter example?
Sigmoid itself is not a convex function (see this) and square loss based on sigmoid, such as $ \left( A - \frac{1}{1 + e^{-z}} \right)^2 $ is not convex.

Simply plotting the square loss function with A = 5 shows that it's convex for x > 0 and concave, otherwise



If you want mathematical proof, take the second derivative and you'll see that it not strictly positive nor negative.