Most cultivated roses are grafted onto a strong, wild rose root. Sometimes
this root sends up shoots which may eventually kill out the cultivated
variety.
Suckers are light green and their leaves usually look smaller than the
cultivated variety. They will grow from beneath the graft (the bit where
the shoots grow out of the root).
You need to pull (not cut) these off right at the root to get rid of them
totally. You may need to remove some soil to get to them.