On polarizations in invariant theory

Given a reductive algebraic group $G$ and a finite dimensional algebraic $G$-module $V$, we study how close is the algebra of $G$-invariant polynomials on $V^{\oplus n}$ to the subalgebra generated by polarizations of $G$-invariant polynomials on $V$. We address this problem in a more general setting of $G$-actions on arbitrary affine varieties.