# In a model with several parameters, which one should be tuned via cross validation first?

I have a loss function like

\eqalign{ L(U, V, P, Q) = & \alpha_1 (R - U \cdot V^T )^2 + \alpha_2 (D - U \cdot P^T )^2 + \alpha_3 (S - V \cdot Q^T )^2 \\ &+ \lambda_1 (\parallel U \parallel^2 + \parallel V \parallel^2) + \lambda_2 \parallel P \parallel^2 + \lambda_3 \parallel Q \parallel^2 }

where $\alpha_{1,2,3}$ are weights and $\lambda_{1,2,3}$ are regularization parameters.

Now I'd like to find the best parameters via cross validation, but there are two questions:

Question 1. Should I tune the weight parameter first, or regularization parameter first? Does the order matter?

Question 2. Among weight parameters, which one should I tune first, and which value should I fix for the rest?

Because I thought these parameters are dependent on each other, say $\alpha_1$ and $\alpha_2$, if I fix $\alpha_2$ to some value (and I also don't know which value it should be), and then tune $\alpha_1$, how can I make sure the tuned $\alpha_1$ is globally optimum?