Is there any point putting two simlar regularization terms in the objective function

Question

I am trying to implement a dictionary learning based objective function. I have created two models as follows:

1) $||X_{s} - Y_{s}D_{s}||_{F}^{2} + \lambda_{1}(||D_{s}||_{F}^{2} - 1)$

2) $||X_{s} - Y_{s}D_{s}||_{F}^{2} + \lambda_{1}(||D_{s}||_{F}^{2} - 1) + \lambda_{2}||D_{s}||_{F}^{2}$

where the first term is the reconstruction loss and the regularization term puts a constraint on values that $D_{s}$ may assume.

In my opinion, the last term i.e. $\lambda_{2}||D_{s}||_{F}^{2}$ in the second model is extraneous because we have already ensured a similar constraint in the first model using $\lambda_{1}(||D_{s}||_{F}^{2} - 1)$.

Any comments ?

Answer 1

You are right, the optimization problem is equivalent.

$\arg \min ( ||X_s−Y_sD_s||^2_F+λ_1(||D_s||^2_F−1)+λ_2||D_s||^2_F )$ = $\arg \min ( ||X_s−Y_sD_s||^2_F+(λ_1+λ_2)||D_s||^2_F-λ_1 ) $ = $\arg \min ( ||X_s−Y_sD_s||^2_F+(λ_1+λ_2)||D_s||^2_F) $ = $\arg \min ( ||X_s−Y_sD_s||^2_F+\tilde{λ}_1||D_s||^2_F ) $