EssayproSolutions

Exercice 2 : Universal Consistency We still consider the binary classification problem on $\mathcal{X}$, and we denote f​n​:F(X,Y) a classifier. We say that it is universally consistent in probability, if $\forall \delta >0$, supP​P(∣∣​LP​(f​n​)−LP​(fP∗​)∣∣​>δ)n→∞⟶​0, where we precise in index the probability distribution (be careful, for the convergence in probability and the computations of expected risk). We suppose that $\mathcal{X}$ is finite : $\mathrm{Card}(\mathcal{X})=K$. 1. What is the cardinality of $\mathcal{F}(\mathcal{X},\mathcal{Y})$ ? 2. Recall the risk bound proved in the class for the ERM f​n, ERM ​ with respect to $\mathcal{F}(\mathcal{X},\mathcal{Y})$. Can we dedude that f​n,ERM​ is universally consistent? 3. We now suppose that K=Kn​ depends on the sample size. Prove that if Kn​ is sub-linear in $n$, then f​n,ERM​ est universally consistent.