**Computational learning theory**is the analysis of computational complexity of machine learning algorithms. It is the intersection oftheory of computation and machine learning.

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 the samples that have never been previously seen 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
- Cryptographic - One-way functions exist.

There are several different approaches to computational learning theory. These differences are based on making 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:

- 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. M. Gold.
- Online machine learning, from the work of Nick Littlestone.

Computational learning theory has led to several practical algorithms. For example, PAC theory inspired boosting, VC theory led to support vector machines, and Bayesian inference led to belief networks (by Judea Pearl).

https://en.wikipedia.org/wiki/Computational_learning_theory