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b/1year/3trimester/Machine Learning, Statistical Learning, Deep Learning and Artificial Intelligence/Machine Learning/lectures/lecture7.tex @@ -1,6 +1,36 @@ \documentclass[../main.tex]{subfiles} \begin{document} \chapter{Lecture 7 - 07-04-2020} - - +Bounding statistical risk of a predictor\\\ +Design a learning algorithm that predict with small statistical risk\\ +$$ +(D,\ell) \qquad \ell_d(h) = \barra{E}\left[ \, \ell (y), h(x) \, \right] +$$ +were $D$ is unknown +$$ +\ell(y, \hat{y}) \in [0,1] \quad \forall y, \hat{y} \in Y +$$ +We cannot compute statistical risk of all predictor.\\ +We assume statistical loss is bounded so between 0 and 1. Not true for all +losses (like logarithmic ).\\ +Before design a learning algorithm with lowest risk, How can we estimate +risk?\\ +We can use test error $\rightarrow$ way to measure performances of a predictor h. +We want to link test error and risk. +\\ +Test set $S' = \{ (x'_1, y'_1) ...(x'_n,y'_n) \}$ is a random sample from $D$ +\\ +How can we use this assumption?\\ +Go back to the definition of test error\\ +\\ +\red{ Sample mean (IT: Media campionaria)}\\ +$$ +\hat{\ell}_s(h) = \frac{1}{n} \cdot \sum_{t=1}^{n} \ell (\hat{y}_t,h(x'_t)) +$$ +i can look at this as a random variable +\col{$\ell(y'_t,h(x'_t))$}{Blue} +\\ +$$ +\barra{E} \left[ \, \ell (y'_t, h(x'_t)) \right] = \ell_D(h) \longrightarrow \red{risk} +$$ \end{document} \ No newline at end of file