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In most of the gradient search examples, the update to weights are done by subtracting the derivative of MSE.

Can we have an example, where we did not use derivative of MSE, but used derivative of some other loss function?

If so, please show the Python code.

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Yes we have. MSE loss function is usually used for regression problems. We can have different loss functions according to our needs. Follow the article below. https://towardsdatascience.com/common-loss-functions-in-machine-learning-46af0ffc4d23

If you wish to learn as to what does gradient does in the algorithm you can visit this article: https://towardsdatascience.com/wondering-why-do-you-subtract-gradient-in-a-gradient-descent-algorithm-9b5aabdf8150

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