Learning the Hypotheses Space from data Part II: Convergence and Feasibility

In part $\mathit{\text{I}}$ we proposed a structure for a general Hypotheses Space $\mathcal{H}$, the Learning Space $\mathbb{L}(\mathcal{H})$, which can be employed to avoid $\mathit{\text{overfitting}}$ when estimating in a complex space with relative shortage of examples. Also, we presented the U-curve property, which can be taken advantage of in order to select a Hypotheses Space without exhaustively searching $\mathbb{L}(\mathcal{H})$. In this paper, we carry further our agenda, by showing the consistency of a model selection framework based on Learning Spaces, in which one selects from data the Hypotheses Space on which to learn. The method developed in this paper adds to the state-of-the-art in model selection, by extending Vapnik-Chervonenkis Theory to $\mathit{\text{random}}$ Hypotheses Spaces, i.e., Hypotheses Spaces learned from data. In this framework, one estimates a random subspace $\hat{$\mathcal{M}$} \in $\mathbb{L}$($\mathcal{H}$)$ which converges with probability one to a target Hypotheses Space ${\mathcal{M}}^{\star }\in \mathbb{L}(\mathcal{H})$ with desired properties. As the convergence implies asymptotic unbiased estimators, we have a consistent framework for model selection, showing that it is feasible to learn the Hypotheses Space from data. Furthermore, we show that the generalization errors of learning on $\widehat{\mathcal{M}}$ are lesser than those we commit when learning on $\mathcal{H}$, so it is more efficient to learn on a subspace learned from data.