Arg Min in LASSO Regression

I am trying to wrap my head around Arg Min, and I think I may have grasped it, so I want to discuss the interpretation in a specific example from a LASSO regression model $$\hat{\beta_h} = \arg \min_{\beta_h} \quad RSS + \lambda \sum_{j=1}^{247} \vert \beta_h \vert$$ From what I understand, this example says that $$\hat{\beta_h}$$, should be equal to the $${\beta_h}$$ which minimizes the function $$\quad RSS + \lambda \sum_{j=1}^{247} \vert \beta_h \vert$$.

I am not sure if this interpretation is correct.

Bu this brings another questions, shouldn't also $$\lambda$$ should be chosen, together with $$\beta_h$$ to minimize $$\quad RSS + \lambda \sum_{j=1}^{247} \vert \beta_h \vert$$. Where does this happen?

Thanks.

The sum $$\sum_{j=1}^n |\beta_j|$$ imposes a restriction to your parametric set in the form of $$n$$-dimensional diamond. Therefore, the arg min of your function, i.e., the minimal $$\beta$$, should be inside this diamond. Clearly, the borders of this diamond are determined by $$\lambda$$, and if the arg min is further away from $$0$$, your answer may be very affected by the size of $$\lambda$$. Therefore, $$\lambda$$ is usually varied on a set of predetermined values, and then the optimal $$\lambda$$ is chosen according to some criteria, e.g., $$\lambda$$ that minimizes the mean squared error.