Computational learning theory

Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible. The algorithm takes these previously labeled samples and uses them to induce a classifier. This classifier is a function that assigns labels to samples, including samples that have not been seen previously by the algorithm. The goal of the supervised learning algorithm is to optimize some measure of performance such as minimizing the number of mistakes made on new samples.

In addition to performance bounds, computational learning theory studies the time complexity and feasibility of learning. In computational learning theory, a computation is considered feasible if it can be done in polynomial time. There are two kinds of time complexity results:

• Positive results – Showing that a certain class of functions is learnable in polynomial time.
• Negative results – Showing that certain classes cannot be learned in polynomial time.

Negative results often rely on commonly believed, but yet unproven assumptions, such as:

• Computational complexity – P ≠ NP (the P versus NP problem);
• Cryptographic – One-way functions exist.

There are several different approaches to computational learning theory based on making different assumptions about the inference principles used to generalize from limited data. This includes different definitions of probability (see frequency probability, Bayesian probability) and different assumptions on the generation of samples. The different approaches include:

• Exact learning, proposed by Dana Angluin;
• Probably approximately correct learning (PAC learning), proposed by Leslie Valiant;
• VC theory, proposed by Vladimir Vapnik and Alexey Chervonenkis;
• Bayesian inference;
• Algorithmic learning theory, from the work of E. Mark Gold;
• Online machine learning, from the work of Nick Littlestone.

While its primary goal is to understand learning abstractly, computational learning theory has led to the development of practical algorithms. For example, PAC theory inspired boosting, VC theory led to support vector machines, and Bayesian inference led to belief networks.

• Grammar induction
• Information theory
• Stability (learning theory)
• Error Tolerance (PAC learning)

• Basics of Bayesian inference