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Is the cost function of a neural network with only linear activation functions a convex function with respect to its parameters? If it is, how to prove it?

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    $\begingroup$ Linear functions are closed under composition; linear regression is convex under some assumptions. $\endgroup$ – Sycorax Mar 2 at 2:11
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I'm opening up the statement made in the comments.

A neural network with linear activation and many layers is equivalent to a neural network with one layer. For example:

$$W_2(W_1x+b_1)+b_2=Wx+b$$

So, in the end, your equation is $Wx+b=y$, i.e. just a linear regression. The cost function is already convex, because it is norm-squared of an affine function. Also, more strongly, the problem is a convex problem.

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