Lecture 6. PAC Learning Theory#

Probably Approximately Correct (PAC) Learning Theory:

Definition: PAC learning is a framework in computational learning theory that studies the feasibility of learning a concept from a given class of concepts, based on a set of training examples. A concept class is said to be PAC-learnable if, given a sufficiently large set of random samples, a learner can approximate the concept with high accuracy, with high probability.

Key Ideas:

  1. Probably: Refers to the algorithm’s ability to produce a hypothesis that is “probably” (i.e., with high probability) close to the true concept.

  2. Approximately: Indicates that the algorithm’s output is allowed to be “approximately” (i.e., within some error tolerance) correct.

  3. Efficiency: A concept class is PAC-learnable if there exists an algorithm that can learn the class in polynomial time based on the size of the examples, the error tolerance, and the confidence parameter.

Importance of PAC Learning Theory:

  1. Foundation for Machine Learning Theory: PAC learning provides a rigorous theoretical foundation for understanding the learnability of concepts, bridging computer science and statistics.

  2. Performance Guarantees: It provides performance guarantees for learning algorithms under certain conditions, giving insights into how many samples are needed to learn a concept to a given degree of accuracy.

  3. Model Evaluation: PAC theory allows for the comparison of different learning algorithms in a formal way, offering a basis for understanding their relative strengths and weaknesses.

  4. Generalization Bounds: It provides bounds on the generalization error, helping to understand how well a learned model will perform on unseen data.

  5. Guidance for Algorithm Design: The theoretical insights from PAC learning have inspired the design of new algorithms and models in machine learning.

  6. Distribution-free Assumptions: PAC learning makes minimal assumptions about the distribution of the data, making it a robust framework for analyzing the learnability across various data distributions.

In summary, PAC learning theory is a foundational framework in machine learning that provides a rigorous understanding of the learnability of concepts. Its importance lies in offering performance guarantees, guiding algorithm design, and providing insights into the generalization capabilities of models.

Lecture 6: PAC Learning Theory (COMP90051 Statistical Machine Learning)

  1. Introduction:

    • Focus on Excess Risk and its decomposition into Estimation vs. Approximation.

    • Introduction to the concept of Bayes risk, which represents the irreducible error.

    • Introduction to Probably Approximately Correct (PAC) learning.

  2. Generalisation and Model Complexity:

    • Previous theories focused on asymptotic notions (consistency, efficiency).

    • The need for finite sample theory is emphasized.

    • Model complexity and its relation to test error is discussed.

  3. PAC Learning:

    • This is the bedrock of ML theory in computer science.

    • In supervised binary classification, the goal is to learn a function that classifies data into one of two labels.

    • Test error is bounded in relation to the Bayes risk.

  4. Decomposed Risk:

    • Risk is decomposed into:

      • Estimation Error (Good)

      • Approximation Error (Ugly)

      • Excess Risk (Bad)

    • The trade-off between simple models (may underfit) and complex models (may overfit) is discussed.

  5. Bayes Risk:

    • Represents the best possible risk.

    • Depends on distribution and loss function.

    • Bayes classifier achieves the Bayes risk.

  6. Bounding True Risk:

    • The concentration inequality is introduced to bound the risk.

    • Hoeffding’s inequality is presented, which bounds how far a mean is likely to be from an expectation.

    • This provides a bound on true risk based on empirical risk.

  7. Uniform Deviation Bounds:

    • There’s a need to uniformly cover the deviation over a family of functions.

    • This is crucial for analyzing risks of models learned from specific learners.

    • The goal is to bound the worst deviation over an entire function class.

  8. Finite Function Classes:

    • The Union Bound concept is introduced.

    • A uniform deviation bound over any finite class or distribution is presented.

  9. Discussion:

    • Hoeffding’s inequality and its implications are discussed.

    • The potential limitations of the union bound are presented.

    • The upcoming topic, VC theory, is hinted at.

  10. Summary:

    • The lecture delves deep into PAC learning, emphasizing the importance of understanding risk, its decomposition, and the tools to bound them.