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I am working on my MSc. Statistic which is on the Penalized Logistic Regression with the LASSO penalty.

I am trying to understand the difference in two objective functions:

argmin {$\frac{1}{n}$ $\sum_{i=1}^n (Y_i-{\beta_0}-\sum_{j=1}^p {\beta_j}X_ij)^2+{\lambda}||{\beta}||_1$}

argmin { $\sum_{i=1}^n (Y_i-{\beta_0}-\sum_{j=1}^p {\beta_j}X_ij)^2+{\lambda}||{\beta}||_1$}

as you can see in the first formula we have $\frac{1}{n}$ while in the 2nd we don't when $n$ represent the sample size. Does anyone know why is this?

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Both formulas are equivalent, since the $argmin$ will be the same regardless of whether you divide by n or not. You could also add an arbitrary constant the the expression would remain equivalent. For example, 3 is less than 5, so for every $a>0$ and any number $b$, $\frac{3}{a} + b < \frac{5}{a} + b$ In short, stay with the one you sounds more intuitive to you

For a similar reason, we sometimes use "sum of squares" instead of mean-squared error when comparing regression models, as the only difference is in scale

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    $\begingroup$ One thing to note is that the scale of $\lambda$ will change based on which expression is used, but as long as an appropriately scaled grid is used to choose $\lambda$, the coefficient values at the optimum will be the same. $\endgroup$
    – Noah
    Jun 20, 2019 at 15:39

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