E.g. is it possible to solve the XOR problem without backpropagation. If so, what would a solution look like?
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Purpose of Backpropagation is to optimize the bias of each hidden layer and the weights on the connecting neurons. So without backpropagation you can solve the problem but you won't be getting correct results. following link may be useful: https://datascience.stackexchange.com/questions/11589/creating-neural-net-for-xor-function

Harshit Mehta
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How does is it solve the problem if the result is incorrect? – Emil Jul 29 '17 at 08:36
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What i want to imply is that suppose you have accuracy as 70% without back propagation but this accuracy can be increased to 90 plus accuracy by including backpropagation – Harshit Mehta Jul 29 '17 at 14:38