Motivation and the PAC Framework
When you train a model on a finite dataset, how can you be confident that it will perform well on new, unseen examples? This question lies at the heart of machine learning. In real-world applications, you rarely have access to the entire population of data. Instead, you collect a sample, train your model, and hope it generalizes beyond what it has seen. However, without a theoretical foundation, you cannot be sure that a model's performance on training data will translate to future data. Overfitting is always a risk: a model might simply memorize the training data, failing to capture the underlying patterns. To address this uncertainty, you need formal guarantees about a model's ability to generalize. This is where the framework of Probably Approximately Correct (PAC) learning becomes essential. PAC learning provides a rigorous way to quantify the reliability of learning algorithms, offering answers to the question: under what conditions, and with what confidence, can you trust that your model will perform well on unseen data?
The Probably Approximately Correct (PAC) learning framework formalizes the process of learning from data. In PAC learning, a learning algorithm is said to be probably approximately correct if, for any small error rate (ε, "approximately correct") and any small failure probability (δ, "probably"), the algorithm can, with probability at least 1−δ, find a hypothesis whose error on unseen data is at most ε, provided it is given enough training examples.
"Probably" refers to the fact that learning from a finite sample can never guarantee perfect performance every time—randomness in sampling means there is always a small chance of failure. "Approximately correct" acknowledges that perfect accuracy on unseen data is unrealistic; instead, you aim for error below a small threshold ε. PAC learning guarantees that, with high probability (at least 1−δ), your model's error will be acceptably small (at most ε), as long as you use enough data.
A hypothesis class H is PAC-learnable if there exists a learning algorithm such that, for any 0<ε,δ<1, and for any distribution over the data, the algorithm will, with probability at least 1−δ (over the random choice of training data), output a hypothesis h∈H whose true error is at most ε, given a sufficiently large number of training examples (sample complexity depending on ε and δ).
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When you train a model on a finite dataset, how can you be confident that it will perform well on new, unseen examples? This question lies at the heart of machine learning. In real-world applications, you rarely have access to the entire population of data. Instead, you collect a sample, train your model, and hope it generalizes beyond what it has seen. However, without a theoretical foundation, you cannot be sure that a model's performance on training data will translate to future data. Overfitting is always a risk: a model might simply memorize the training data, failing to capture the underlying patterns. To address this uncertainty, you need formal guarantees about a model's ability to generalize. This is where the framework of Probably Approximately Correct (PAC) learning becomes essential. PAC learning provides a rigorous way to quantify the reliability of learning algorithms, offering answers to the question: under what conditions, and with what confidence, can you trust that your model will perform well on unseen data?
The Probably Approximately Correct (PAC) learning framework formalizes the process of learning from data. In PAC learning, a learning algorithm is said to be probably approximately correct if, for any small error rate (ε, "approximately correct") and any small failure probability (δ, "probably"), the algorithm can, with probability at least 1−δ, find a hypothesis whose error on unseen data is at most ε, provided it is given enough training examples.
"Probably" refers to the fact that learning from a finite sample can never guarantee perfect performance every time—randomness in sampling means there is always a small chance of failure. "Approximately correct" acknowledges that perfect accuracy on unseen data is unrealistic; instead, you aim for error below a small threshold ε. PAC learning guarantees that, with high probability (at least 1−δ), your model's error will be acceptably small (at most ε), as long as you use enough data.
A hypothesis class H is PAC-learnable if there exists a learning algorithm such that, for any 0<ε,δ<1, and for any distribution over the data, the algorithm will, with probability at least 1−δ (over the random choice of training data), output a hypothesis h∈H whose true error is at most ε, given a sufficiently large number of training examples (sample complexity depending on ε and δ).
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