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Is the PAC-learning theory actually used in every day machine learning? Or is this something that you learn at university and don't really need unless you do research on algorithms and need to provide mathematical proofs?

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As you know the PAC learnability is a concept in theoretical machine learning. Hence, it's a fundamental concept and mostly used in researches and proving some theorems. However, you can use from the bounds to estimate the size of training data and the accuracy of your learning methods in the everyday machine learning!

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