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goes down \qquad (if not underfitting) +\end{itemize} +ERM with $|H| < \infty$ +\\ +$A$ \qquad $H$ such that $\forall s$ $A(S) \in H$ +\\\\ +We controlled this event: +$$ +\forall h \in H \qquad \hat{\ell}_S(h) - \ell_D \left(h\right) \quad \leq \quad \sqrt[]{\frac{2}{m}\, \ln \, \frac{2\,|H|}{\delta}} +\qquad +\textit{ with probability at least $1-\delta$} +$$\\ +We assure that training error is a good proxy for the true risk. +\begin{figure}[h] + \centering + \includegraphics[width=0.5\linewidth]{../img/lez8-img2.JPG} + \caption{Example} + %\label{fig:} +\end{figure}\\ +\bred{If I do the empirical way(for a specific training set) i get something different.} +\\ +\section{The problem of estimating risk in practise} +Test error is a good estimate for a risk, provided $\frac{1}{\sqrt[]{n}}$ is small\\\\ +It’s usually good to take a small test set.\\ +$80/20$ big data, if low data better to look at good estimate...=??\\ +Usually small test set is still ok\\\\ +Typically we are not given the predictor, instead we start from a learning +algorithm. It’s true that A has parameter (how many nodes for a tree classifier +for example).\\ +In general, if I have parameters then i get a set of different classifier +Imaging Algorithm that has parameter $\theta$: +$$ +A \qquad \{ A_{\theta}: \theta \in \Theta \} \qquad +A_{\theta} \left(S\right) = \hat{h}_S +$$ +$\barra{E} \left[ \ell_D \left( A_\theta \left(S \right) \right) \right] +$ \qquad \bred{typically averaged over $S$ of size $m$ for fixed $m$} +\\\\In general you may also be interest looking at the best choice of parameter for +your algorithm: +Suppose now that i want to look at mini sing the risk with the respect to the +choice of the parameter\\\\ +$\barra{E} \left[ \, min \, \ell_D \left( A_\theta \left(S \right) \right) \right] +$ \quad i want to choose parameter that minimise risk running algorithm on training set. +\\\\ +I would like to choose best possible value for my parameter k in the k-NN and +i want to estimate the risk.\\ +$\theta$ can be a set of parameters (so more than 1 parameter), so theta are +the \textbf{hyper parameters of A}.\\ +In general this parameter are the choices that algorithm that can make: +parameter that define a classifier (example nodes and test in the internal +nodes and label in the leaves)\\\\ +\bred{Hyper parameters} are not determined by the training set $\rightarrow$ chosen +before the training set.\\ +I fix $k$ for $\knn$ and I choose an upper bound of number of nodes after the +training set is given.\\ +So, some parameters are given after training set is given and some parameter +fixed before.\\ +Now the idea is that parameter are determined after training set is given, while +hyper parameters are given before training set to get then a predictor.\\ +I have a family of algorithms, so I choose a hyper parameters to get a one +algorithm.\\ +Most algorithm are given as family. We have first decide hyper parameters not +determined on the training set.\\ +One way to move is to take you dataset and the split that in 3 parts: +\begin{itemize} +\item Training set +\item Development (or validation) test +\item Test sets +\end{itemize} +There should not be a leak of information between test and train and +development.\\ +We get family of algorithm, train with train set and then use dev set to test the +parameter and choice the parameters. +Once i found the parameter I retrain the algorithm with a part of train and dev +set to being then tested. +\\ +\bred{Development set is like a fake test set } $\rightarrow$ it usefull to choose parameters. +\\ +Algorithm steps: +\begin{enumerate} +\item Train $A_\theta$ on training set for each $\theta \in \Theta $ (\bred{grid search}) +\item Find $\theta$ such that $\hat{h}= A_{\hat{\theta}}\left(S\right)$ \, minimise developement error +\item Train $A_{\hat{\theta}}$ on training $+$ development set +\item Test resulting predictor on test set +\end{enumerate} +(There's theory about this but it's difficult and we are not going to do that) +\\\\ +It’s heuristic and kinda simple to do. This technics work for every learning +algorithm.\\ +One parameter: grid on this\\ +Two parameters ecc..\\ +\textbf{It’s quadratic!}\\ +\bred{Good learning algorithm should have small number of hyperparameters.} + +\section{Cross-validation} + +Solving an easier problem: suppose you have a dataset and what you do is +that you can choose training set and test set.\\ +Algorithm $A$ with no hyper parameters and we would like to check how good is +the predictor: how good can $A$ be? $A$ is good if the predictor that generate has +low risk.\\ +I split dataset in training and test set. +\begin{figure}[h] + \centering + \includegraphics[width=0.5\linewidth]{../img/lez8-img3.JPG} + \caption{Splitting test and training set} + %\label{fig:} +\end{figure} +$$ +h = A\left(A\right) \qquad \hat{\ell}_{S'}\left(h\right) \approx \ell_D\left(h\right) +$$ +I can use Cross-validation! (CV). It helps estimate the risk. +$$ +\bred{CV:} \qquad \barra{E}\left[ \ell_D \left(A\left(S \right) \right) \right] +$$ +I would like to average .... \\ +How do i do this estimate? Super easy, take my data assuming CV as +parameter not of the algorithm A, but intrinsic to cross validation.\\ +Parameter k-fold CV typically $k = 5$ or $10$.\\ +What do you do ? If i take a specific training set, i got a ruff estimate of A. +I shouldn’t split one but several time.\\ +There are different way but he give us another:\\ +\textbf{split data in k folds randomly!} +\begin{figure}[h] + \centering + \includegraphics[width=0.6\linewidth]{../img/lez8-img4.JPG} + \caption{K-folds} + %\label{fig:} +\end{figure}\\ +For each $k$ we define $S^{(k)} + \qquad \bred{take out k-st D}$ +$$ +S^{(k)} = S\backslash D_k +$$ +$$ +S^{(1)} = D_2 \cup D_3 \cup ...\cup D_k +$$ +$$ +S^{(2)} = D_1 \cup D_3 \cup ... \cup D_k +$$ +For each $k-1, ... k$ \bred{folds} \qquad $S^{(k)}$ training part and $D_k$ test part +$$ +h_K = A \left( S^{(k)} \right) \qquad \hat{\ell}_{DK} \left(h_K\right) = \frac{k}{m} \, \sum_{ (x,y) \in D_k}{} \ell \left( y, h_K\left( x \right) \right ) +$$ +Repeat the procedure for $k = 1 ... k$ \quad get $h_1, ... , h_k$ +\\ +$$ +Compute \qquad \frac{1}{k} \cdot \sum_{k=1}^{K} \hat{\ell}_{DK} \left(h_k\right) +$$ +where $CV$ is $\barra{E}\left[ \ell_D \left(A\left(S \right) \right) \right]$ and this is called ..... ---MANCA --- Estimate +\\\\ +It’s used a lot!\\ +You get data from internet, then what to do? I want to try an algorithm, try $\knn$ so you do Cross validation and will give you the risk.\\ +In some other cases you get a splitted dataset in training and testing set. +You don’t use CV since the dataset is already splitted in training and test. + +\section{Nested cross validation} +The use of CV to solve the hyper parameters choice problem. +If you are given test and training set you can solve splitting in training set and +dev set.\\ +Suppose not given train and test, you can assign arbitrary .....\\ +Now the idea: you give me a way to optimise splitting training set and test set +and then avoiding using a ...\\ +\textbf{This is what cross validation is doing.} +\begin{figure}[h] + \centering + \includegraphics[width=0.8\linewidth]{../img/lez8-img5.JPG} + \caption{Nested Cross Validation} + %\label{fig:} +\end{figure}\\ +$ \{ A_\theta : \theta \in \Theta \}$ +which i run in each fold? +\\ +The idea is \bred{running interval CV for the folds}\\ +I have fold $D_1$ and $S(1)$ and i perform external validation, then i run internal +CV in $S(1)$.\\ +On each training part of the external CV, run a internal CV for each $A_\theta$. +\\When $\theta \in $ grid ($\Theta$): +\begin{itemize} +\item I have a CV-estimate and for each $A_\theta$ pick the $\theta$ with best CV-estimate +\item Run $A_{\hat{\theta}}$ on the entire training part of current external fold +\end{itemize} +\textbf{Basically what I’m choosing the best hyper parameter on each fold of the +external CV.}\\ +External CV is not testing $A_{\hat{\theta}}$ for a given $\hat{\theta} \in \Theta$ +\\ +I am not measuring a goodness of predictor generate by algorithm for given +value of hyper parameters but what I’m estimating is the average risk of the +predictor output by learning algorithm when hyper parameters are optimise on +the training set. This optimisation on training set is done into a internal CV. To +avoid be depend i run and external CV.\\ +Many platform like sklearn allow you to do that in two lines of code. So i can +specify the grid, predictor, number of falls for internal and external. +It took a bit but that’s it in in two lines of code\\ +Cross validation can be done in every order of $D_1$ or $D_2$ or $D_k$. So doesn’t +matter the order we start fold D2 or D1. \end{document} \ No newline at end of file