We have finally arrived at the last post of our series of the proof that linear regression indeed is a sharp learner. Recall that in the first post we began by motivating linear regression as a problem on predicting house prices and quickly came to understand there was a beautiful way to frame this probem abstractly: given any set of features $\mathfrak{X}$ and fd Euclidean space of labels $\mathfrak{y}$, as well as dataspace $\mathfrak{D}$ satisfying the separation condition for a finite dimensional hypothesis space $\mathfrak{H}\subset \mathfrak{y}^{\mathfrak{X}}$, is it possible to find a map $$h:\mathfrak{D} \longrightarrow \mathfrak{H}$$ such that $c(\Delta, h_\Delta)=\min_{h \in \mathfrak{H}} c(\Delta,h)$ where $$ c(\Delta, h)=\sum_{(x,y)\in \Delta}\vert \vert y-h(x)\vert\vert^2 $$

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