From 97c8156e45eb6bc8bfa6aec8e72e58e4bf9da2f8 Mon Sep 17 00:00:00 2001 From: Yawei Li Date: Wed, 23 Oct 2024 11:44:55 +0200 Subject: [PATCH] add faster solution to sol_advriskmin_1 (b) --- exercises-pdf/ex_advriskmin_1.pdf | Bin 136959 -> 136956 bytes exercises-pdf/ex_advriskmin_2.pdf | Bin 181007 -> 181011 bytes exercises-pdf/ex_advriskmin_3.pdf | Bin 169541 -> 175489 bytes exercises-pdf/ic_advriskmin_1.pdf | Bin 93093 -> 93093 bytes exercises-pdf/ic_sol_advriskmin_1.pdf | Bin 137043 -> 137043 bytes exercises-pdf/sol_advriskmin_1.pdf | Bin 182992 -> 185878 bytes exercises-pdf/sol_advriskmin_2.pdf | Bin 239677 -> 245632 bytes exercises-pdf/sol_advriskmin_3.pdf | Bin 202213 -> 202205 bytes .../ex_rnw/sol_generalized_l2_loss.Rnw | 29 +++++++++++++++++- .../advriskmin/figure/unnamed-chunk-2-1.pdf | Bin 6155 -> 6155 bytes .../advriskmin/figure/unnamed-chunk-2-2.pdf | Bin 6500 -> 6500 bytes 11 files changed, 28 insertions(+), 1 deletion(-) diff --git a/exercises-pdf/ex_advriskmin_1.pdf 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\textbf{Recommended solution:} +We need to use the fact that $y,y^{(1)},\ldots,y^{(n)}$ are i.i.d. and the arithmetric mean is an unbiased estimator. That being said, +\begin{align*} + \E_{xy}[m(y)] - \E_{xy}\left[\frac{1}{n} \sum_{i=1}^n m(\yi)\right] = 0. +\end{align*} + +Therefore, + +\begin{align*} +% + \risk_L\left(\fh \right) + &= \E_{xy} [\Lxy] \\ + &= \E_{xy} [ \big(m(y)-m(\fxh)\big)^2 ] \\ + &= \E_{xy} \left[ \left( m(y) - \frac1n \sum_{i=1}^n m(\yi) \right)^2 \right] \\ + &= \E_{xy} \left[ \left( m(y) - \frac{1}{n} \sum_{i=1}^n m(\yi) - \left( \E_{xy}[m(y)] - \E_{xy}\left[\frac{1}{n} \sum_{i=1}^n m(\yi)\right] \right) \right)^2 \right] \\ + &= \E_{xy} \left[ \left( m(y) - \frac{1}{n} \sum_{i=1}^n m(\yi) - \E_{xy}\left[m(y) - \frac{1}{n} \sum_{i=1}^n m(\yi)\right] \right)^2 \right] \\ + &= \var\left( m(y) - \frac{1}{n} \sum_{i=1}^n m(\yi) \right) \\ + &= \var(m(y)) + \var\left(\frac{1}{n} \sum_{i=1}^n m(\yi) \right) \\ + &= \var(m(y)) + \frac{1}{n^2}\var(\sum_{i=1}^n m(\yi)) \\ + &= \var(m(y)) + \frac{1}{n^2} \cdot n \var(m(\yi)) \\ + &= \var(m(y)) + \frac{1}{n^2} \cdot n \var(m(y)) \\ + &= \left(1 + \frac{1}{n} \right) \var(m(y)) \\ +\end{align*} + + +\textbf{Alternative solution:} + +Here we provide an alternative solution to (b). 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