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chapter1.tex
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chapter1.tex
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\section{Introduction (Lec. 1)}
\subsection{The Well-specified Case}
Given a collection of $n$ samples $\{x_i\}_{i=1}^n$ sampled from a fixed distribution $\mathbb{P}$. Let $f^*$ be the ``true'' estimator that minimizes some loss $\ell(x; f)$, i.e., $f^* := \arg \min_{f\in\mathcal{F}} \mathbb{E}_x\ell(x_i; f)$.
\begin{definition}[Risk] The estimation of losses is called \textit{risk}:
\begin{itemize}
\setlength\itemsep{-0.5em}
\item Empirical risk: $$R_n(f) := \frac{1}{n} \sum_{i=1}^n \ell(x_i; f)$$
\item Population risk: $$R(f) := \mathbb{E}_{x \sim \mathbb{P}} \ell(x; f)$$
\end{itemize}
\end{definition}
\textbf{Remark}: Please note that the notation in our course is slightly different from the MW Chapter 4, where it uses a more burdensome notation as $R(f, f^*)$. Here, the $f^*$ is omitted since it is fixed in a problem.
\begin{definition}[Excess risk]
The excess risk is defined as $$\mathcal{E}_n(\hat{f}_n, f^*) := R(\hat{f}_n) - R(f^*).$$
\end{definition}
\begin{theorem}[Risk decomposition]
The excess risk $R(\hat{f}) - R(f^*)$ can be decomposed into
\begin{align*} = \underbrace{R(\hat{f}) - R_n(\hat{f})}_{T_1} + \underbrace{ R_n(\hat{f}) - R_n(f^*)}_{T_3} + \underbrace{R_n(f^*) - R(f^*)}_{T_2}
\end{align*}
\end{theorem}
\textbf{Remark}: The following figure illustrate the risk decomposition. In the figure, the upper and lower layers represent the empirical risk and the population risk, respectively.
\begin{center}
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-0.7,xscale=0.75]
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%Shape: Parallelogram [id:dp5659534429170581]
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%Straight Lines [id:da12727382381339924]
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% Text Node
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% Text Node
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% Text Node
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% Text Node
\draw (219,204.4) node [anchor=north west][inner sep=0.75pt] {$R_{n}(\hat{f}_{n})$};
% Text Node
\draw (402.5,176.4) node [anchor=north west][inner sep=0.75pt] {$R_{n}\left( f^{*}\right)$};
% Text Node
\draw (402.5,349.4) node [anchor=north west][inner sep=0.75pt] {$R\left( f^{*}\right)$};
% Text Node
\draw (308.25,362.4) node [anchor=north west][inner sep=0.75pt] [font=\large] {$?$};
\draw (190,282.4) node [anchor=north west][inner sep=0.75pt] {$T_{1}$};
% Text Node
\draw (376,255.4) node [anchor=north west][inner sep=0.75pt] {$T_{2}$};
% Text Node
\draw (296,162.4) node [anchor=north west][inner sep=0.75pt] {$T_{3}$};
\end{tikzpicture}
\vspace{1em}
\caption{\textbf{Figure}: Risk decomposition (well-specified)}
\end{center}
\subsection{The Mis-specification Case}
We call the problem mis-specification, when $f^* \not\in \mathcal{F}$.
\begin{theorem}[Bias-variance trade-off]
The excess risk $R(\hat{f}) - R(f^*)$ can be decomposed into
\begin{align*}
R(\widehat{f}_{n})-R\left(f^{\star}\right)=\underbrace{R(\widehat{f}_{n})-R\left(f^{\star, \mathcal{F}}\right)}_{\text {"Variance" }}+\underbrace{R\left(f^{\star, \mathcal{F}}\right)-R\left(f^{\star}\right)}_{\text {"Bias" }=\text { "approx error" }}
\end{align*}
where $f^{\star, \mathcal{F}} = \arg\min_{f\in\mathcal{F}} R(f)$.
\end{theorem}
\textbf{Remark}: The following figure provides some insights.
\begin{center}
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% Text Node
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% Text Node
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% Text Node
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% Text Node
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% Text Node
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% Text Node
\draw (389.5,279.4) node [anchor=north west][inner sep=0.75pt] {$R\left( f^{*}\right)$};
% Text Node
\draw (282.5,124.4) node [anchor=north west][inner sep=0.75pt] {$R_{n}\left( f^{*,\ \mathcal{F}}\right)$};
% Text Node
\draw (264.5,300.4) node [anchor=north west][inner sep=0.75pt] {$R\left( f^{*,\ \mathcal{F}}\right)$};
\draw (170,202.4) node [anchor=north west][inner sep=0.75pt] {$T_{1}$};
% Text Node
\draw (356,185.4) node [anchor=north west][inner sep=0.75pt] {$T_{2}$};
% Text Node
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% Text Node
\draw (459,362.4) node [anchor=north west][inner sep=0.75pt] [color=red ,opacity=1 ] {$\color{red} \mathrm{bias}$};
% Text Node
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% Text Node
\draw (90,212.4) node [anchor=north west][inner sep=0.75pt] [color=blue ,opacity=1 ] {$\color{blue} \mathrm{variance}$};
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\caption{\textbf{Figure}: Bias and variance under mis-specification}
\end{center}