I was going through this article about regularization, which says:

Note that we do not constrain the y-intercept. We are only concerned with constraining slope angles, not where the line touches the y-axis.

I am not able to get the later statement. I understand that the main goal of regularization is to reduce the chance of overfitting by reducing the sensitivity to small changes in the input data. This is achieved by reducing the parameter values which in turn will reduce the effect of the feature associated with that parameter on the models prediction. Since the intercept doesn't have a feature associated with it, we don't penalize it. ref. But I don't get the visual intuition of slope. Which slope it is talking about in below figures (from very first link)?

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  • 2
    $\begingroup$ it's none of those pictures. the coefficents, betas determine the slope of your output as the corresponding input is changed. if a beta_1 is big then the slope is big, so a small change in the input, x_1 will cause a big change in the output $\endgroup$
    – seanv507
    Nov 30, 2022 at 22:25