  We develop a framework using Hilbert spaces as a proxy to analyze PAC
learning problems with structural properties. We consider a joint Hilbert space
incorporating the relation between the true label and the predictor under a
joint distribution $D$. We demonstrate that agnostic PAC learning with 0-1 loss
is equivalent to an optimization in the Hilbert space domain. With our model,
we revisit the PAC learning problem using methods based on least-squares such
as $\mathcal{L}_2$ polynomial regression and Linial's low-degree algorithm. We
study learning with respect to several hypothesis classes such as half-spaces
and polynomial-approximated classes (i.e., functions approximated by a
fixed-degree polynomial). We prove that (under some distributional assumptions)
such methods obtain generalization error up to $2opt$ with $opt$ being the
optimal error of the class. Hence, we show the tightest bound on generalization
error when $opt\leq 0.2$.
