Does any one else's poodle do this?

My pup is a very picky eater. So picky that she will literally starve herself to the point of throwing up yellow bile (usually in the mornings). I know it is from an empty stomach because there's nothing except for saliva and bile. I know this isn't good for her and would like to minimize if at all possible.

She is healthy otherwise, just stubborn. Does any one else's poodle do this, and if so, how do you deal with it?

Thanks!