This is a research question that I had asked in Math.SE about a month ago, but even after putting a bounty on it, I did not get any answers.
I have two real values functions, where one ($g(w;x):\mathbb{R}^m\rightarrow \mathbb{R}$) asymptotically approximates the other one ($f(w;x):\mathbb{R}^m\rightarrow \mathbb{R}$). $x\in \mathbb{R}^m$ is some known vector, and $w\in \mathbb{R}^m$ is the variable vector.
$f$ can not be evaluated when $\|w\odot x\|_0\geq T$ ($\|w\odot x\|_0$ is the number of non-zeros in the Hadamard (element-wise) product of $x$ and $w$, and $T$ is some integer that is determined by our computational resources.). The approximation is tighter for larger $\|w\odot x\|_0$'s.
Our goal is to $\text{minimize}_w \sum_i f(w;x_i)$.
I would like to analyze the theoretical convergence guarantees when I alternate between different objectives: when $f(w;x_i)$ is easy to evaluate ($\|w\odot x_i\|_0$ is small enough) I add $f(w;x_i)$ to the objective function, and when not, add $g(w;x_i)$.
P.S.: I appreciate any helps or suggestions, and please let me know if this question is not framed appropriately enough for MathOverflow; in particular, please let me know if I need to state the complete question here.
