# PAC learning over continuous functions

I'm wondering if it's possible to use PAC learning to learn a continuous function. For example, if we wanted to learn a probability distribution or a CDF, is it valid to train on some set of m examples and basically then find some hypothesis function h that fits those examples? Basically then we can find the sample complexity as with learning boolean functions, finding what is the minimum sample size we need m that will make the algorithm output an h that will correctly estimate the probability of any input within $$\pm\epsilon$$ for some error parameter $$\epsilon$$

I've seen some papers on using PAC for continuous functions, but they go a bit over my head, and I'm wondering if there's a simpler explanation

• May be of interest to you: en.wikipedia.org/wiki/… (learning a density function in Kolmogorov distance, i.e., learning the CDF to pointwise $\varepsilon$ error) Nov 1, 2022 at 6:26

In PAC learning, you specify the function class a priori. Thus, there might not be a function in your class that fits the sample perfectly. You'll typically minimize some empirical risk, such as $$L_1$$ or $$L_2$$. The classical papers are