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Training error is the prediction error we get applying the model to the same data from which we trained.

Training error is much easier to compute than test error.

Train error is often lower than test error as the model has already seen the training set. It's then going to fit the training set with lower error than it was going to occur on the test set.

And the more we over fit, the harder we fit the data, the lower the training error looks. On the other hand, the test error can be quite a bit higher.

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