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Training order

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Yes.

Consider two entries in your training table, A and B.
You could either train with B then A (i) or train with A then B (ii). Let's say B gives you an error of zero.

(i)
Training with B would leave the net unchanged. Then training with A would adjust the weights.

(ii)
Training with A would adjust the weights, leading to some error in B. Therefore the weights will be adjusted again when you train on B and your net will be different from (i)

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If the data are i.i.d (independent, identically distributed) samples drawn from the domain, then you should be able to present them in any order. If you want to really ensure that you don't learn spurious correlations, present them randomly with a different order each batch iteration.

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