  We show that a $k$-stable set in a finite group can be approximated, up to
given error $\epsilon>0$, by left cosets of a subgroup of index
$\epsilon^{\text{-}O_k(1)}$. This improves the bound in a similar result of
Terry and Wolf on stable arithmetic regularity in finite abelian groups, and
leads to a quantitative account of work of the author, Pillay, and Terry on
stable sets in arbitrary finite groups. We also prove an analogous result for
finite stable sets of small tripling in arbitrary groups, which provides a
quantitative version of recent work by Martin-Pizarro, Palac\'{i}n, and Wolf.
Our proofs use results on VC-dimension, and a finitization of model-theoretic
techniques from stable group theory.
