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There are different types of cost functions like Cross-entropy, absolute error, mean squared error.
When can we expect the cost function to be non-convex? Does this depend upon the type of cost function we choose? Or does this depend upon the Model we choose like when we have higher order polynomial Model?
The cost function is convex if its Second Order Derivative is positive semidefinite (i.e. $\geq0$ ).
But this definition depends on the function with respect to which you take the derivative. This convexity changes when we are talking about Neural Networks, as in that case, our derivatives are taken with respect to the weights.
Refer: Why Cross Entropy is convex, but not necessarily in neural nets and Why RMSE is convex but not necessarily in neural nets
$f(x)$ is convex when $f(a)<f(x)<f(b)$ for every $a<x<b$. Overall, a function with a positive second derivative is convex.
The MSE objective is of the form
$MSE = ∑(y_{true} − 𝑦_{pred})^2$
The second derivative is positive. So, MSE is convex. You can follow this procedure for the rest of the functions.
Here is also another answer to this question.
