# PAC Learnability - Notation

The following is from Understanding Machine Learning: Theory to Algorithm textbook:

Definition of PAC Learnability: A hypothesis class $$\mathcal H$$ is PAC learnable if there exist a function $$m_H : (0, 1)^2 \rightarrow \mathbb{N}$$ and a learning algorithm with the following property: For every $$\epsilon, \delta \in (0, 1)$$, for every distribution $$D$$ over $$X$$, and for every labeling function $$f : X \rightarrow \{0,1\}$$, if the realizable assumption holds with respect to $$\mathcal H,D,f$$ then when running the learning algorithm on $$m \ge m_H(\epsilon,\delta)$$ i.i.d. examples generated by $$D$$ and labeled by $$f$$, the algorithm returns a hypothesis $$h$$ such that, with probability of at least $$1 - \delta$$ (over the choice of the examples), $$L_{(D,f)}(h) \le \epsilon$$.

1) In the function definition $$m_H : (0, 1)^2 \rightarrow \mathbb{N}$$; what does a) 0 and 1 in the bracket, b) the integer 2, and c) $$\rightarrow \mathbb{N}$$ refer to?

• $$m_H:(0,1)^2 \rightarrow \mathbb N$$ is a similar notation to $$f:R^n\rightarrow \mathbb N$$ which means it takes a n-dimensional input consisting of real numbers only. In the case of PAC learning the input is 2 dimensional consisting of numbers between $$0$$ and $$1$$ only which stand for values of $$\epsilon, \delta$$ respectively.
• The integer $$2$$ as explained above is the dimension of the input vector.
• $$\rightarrow \mathbb N$$ means mapped to natural numbers. In case of PAC learning that for each value of $$(\epsilon, \delta)$$ a function $$m_H$$ maps it to natural numbers, simply put $$m_H(\epsilon, \delta) =$$