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While updating weights of the neural network, most of the algorithms use convex optimisation because of the reason that error is a convex function.

My doubt is that whether the convex-ness of error is assumed or real and are there any non-convex error function?

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No, usually the error function is not convex with respect to the weights. Algorithms like (stochastic) gradient descent do not assume convexity -- it's just that you can prove convergence in that case.

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