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\documentclass[../main.tex]{subfiles}
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\begin{document}
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\chapter{Lecture 7 - 07-04-2020}
|
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|
||||
|
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Bounding statistical risk of a predictor\\\
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Design a learning algorithm that predict with small statistical risk\\
|
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$$
|
||||
(D,\ell) \qquad \ell_d(h) = \barra{E}\left[ \, \ell (y), h(x) \, \right]
|
||||
$$
|
||||
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}
|
||||
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Reference in New Issue
Block a user