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Short version: given a linear regression dataset and an integer $K$, what data subset of size $K$ results in linear parameters with the lowest mean squared error on the entire dataset?

Long version: Suppose I have a supervised dataset $D_n := \{x_n, y_n\}_{n=1}^N$ and I want to perform ordinary linear regression by minimizing

$$ L_{OLS, N}(w) = ||Y_N - X_N w||_2^2$$

where $Y_N$ is the matrix formed by stacking $y_n$ as row vectors and $X_N$ is the matrix formed by stacking $x_n$ as row vectors. I use the subscript $N$ to remind us that we used all the data to compute the parameter estimates. Alternatively, I might want to perform ridge linear regression by minimizing

$$ L_{Ridge, N}(w) = ||Y_N - X_N w||_2^2 + c ||w||_2^2$$

We know that the parameters that minimize the above losses are given by

$$w_{OLS, N} = (X_N^T X_N)^{-1} X_N^T Y_N$$

$$w_{Ridge, N} = (X_N^T X_N + c I)^{-1} X_N^T Y_N$$

What I'm curious to know is: if we're forced to choose a subset of only $K < N$ data $D_K := \{(x_k, y_k)\}_{k=1}^K \subset D_N$ and we fit parameters

$$w_{OLS, K} = (X_K^T X_K)^{-1} X_K^T Y_K$$

$$w_{Ridge, K} = (X_K^T X_K + c I)^{-1} X_K^T Y_K$$

what subset of $K$ data results in parameters $w_{OLS, K}$ and $w_{Ridge, K}$ such that $L_{OLS, N}(w)$ and $L_{Ridge, N}(w)$ (respectively) are minimized?

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1 Answer 1

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the squared error of the full data set are: $$SSE=\varepsilon'\varepsilon$$where $\varepsilon=Y-X\hat\beta_K$ and $\hat\beta_K=(X_K'X_K)^{-1}X'_KY_K$

You want to minimize SSE wrt a subset K. Unfortunately the only way to do it is through trying all combinations of K-subsets. It's $\frac{N!}{K!(N-K)!}$, which is a lot of work to say the least.

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  • $\begingroup$ One place to watch out when you try this is if two observations have the same $x$-values but different $y$-values. $\endgroup$
    – Dave
    Nov 18, 2021 at 21:36
  • $\begingroup$ In my setup, I don't get to choose the number of observations. That's a given parameter, and it could be less than the number of effective parameters. $\endgroup$ Nov 18, 2021 at 21:37
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    $\begingroup$ No, not true. You'll get zero errors on those $K$ points, but I want to know about the errors on the full dataset. $\endgroup$ Nov 18, 2021 at 21:38
  • $\begingroup$ Now that I think about it more, I think the above answer isn't correct even when there are more observations than effective number of parameters. $\endgroup$ Nov 18, 2021 at 21:40
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    $\begingroup$ I found Mahoney 2011's "Randomized algorithms for matrices and data" which gives a randomized algorithm with high probability of recovering the correct answer $\endgroup$ Nov 22, 2021 at 21:28

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