# Using the definition of APAC learning and uniform convergence in practice

I am currently studying "Understanding Machine Learning from Theory to Practice" written by Shai Shalev-Shwartz and Shai Ben-David. I want to understand how i can use the Definitions and Results of the Theory he describes in Practice.

Consider the problem of fitting a one dimensional polynomial to data; namely, our goal is to learn a function, $$h : R → R$$ , and as prior knowledge we consider the hypothesis class of polynomials up to degree 10. Our class $$H$$ has VCdim($$H$$)=11 so with the fundamental theorem of statistical learning it is APAC learnable with ERM. If i fix my $$\epsilon,\delta \in (0,1)$$, then if my sample size is equal to $$C_2\frac{11+\log(1/\delta)}{\epsilon^2}$$ i can be sure that with probability of atleast $$1-\delta$$, ERM will output a hypothesis with $$$$L_D(h_s)\leq \min_{h}L_D(h)+\epsilon.$$$$ $$C_2$$ is a constant and $$h_S$$ is the hypotheis of the ERM algorithm. Now in practical Terms, this doesnt tell me anything about the quality of my Model. Because we dont know the underlying Distribution $$D$$ we can't compute $$L_D(h)$$ for any $$h\in H$$. But i can compute $$L_S(h)$$ for every $$h\in H$$ in particular $$L_S(h_S)$$. My intuition is then that we could use the uniform-convergence property of our class $$H$$ (VCdim(H) is finite) to get a bound for how much $$L_D(h_S)-L_S(h_S)$$ differ. With probability of atleast $$1-\delta$$ $$$$L_D(h_S)\leq \min_{h}L_D(h)+\epsilon \leq \min_{h}{L_S(h)+\epsilon}+\epsilon=L_S(h_S)+2\epsilon$$$$ I know my the value of $$\epsilon$$ so i can calculate the above expression. For example with $$\epsilon=0.01$$ and $$L_S(h_S)=0.01$$ i can gurantee that the true error of my hypotheis $$h_S$$ is at most $$\leq 0.03$$ with probability $$1-\delta$$.

Indeed by lemma 4.2, once the training set $$S$$ is $$(\epsilon/2)$$-representative the ERM learning rule is guaranteed to be APAC learnable $$L_D(h_s)\leq \min_{h}L_D(h)+\epsilon$$. However, it's not clear based on what theorem you arrived at $$\min_{h}L_D(h) \leq \min_{h}L_S(h)+\epsilon$$. Since the $$h$$ on both sides may not be the same hypothesis in the hypothesis class under min operator, uniform convergence property doesn't ensure this step at all.