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I am using Tensorflow to train my model. I am routinely saving my model every 10 epochs. I have a limited number of samples to train, so I am augmenting my dataset to make a larger training dataset.

If I need to use my saved model to resume training after a power outage would it be best to resume training using the same dataset or to make a new dataset?

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Your question very much depends on how you're augmenting your dataset. If your augmentation skews the statistical distribution of the underlying dataset then you should resume training with the pre-power outage dataset. Otherwise, you're assuming that your augmentation has not changed the distribution of the dataset.

It is a fairly safe assumption to make (assuming your augmentations do not change the data in an extremely significant way) that you are safe to resume training on a new dataset or the old dataset without significant change in accuracy.