There have been multiple works that try to ascertain explanations for decisions of black box models on particular inputs by perturbing the input or by sampling around it, creating a neighborhood and then fitting a sparse (linear) model (e.g. LIME). Many of these methods are unstable and so more recent work tries to find stable or robust alternatives. However, stable solutions may not accurately represent the behavior of the model around the input. Thus, the question we ask in this paper is are we approximating the local boundary around the input accurately? In particular, are we sampling the right neighborhood so that a linear approximation of the black box is faithful to its true behavior around that input given that the black box can be highly non-linear (viz. deep relu network with many linear pieces). It is difficult to know the correct neighborhood width (or radius) as too small a width can lead to a bad condition number of the inverse covariance matrix of function fitting procedures resulting in unstable predictions, while too large a width may lead to accounting for multiple linear pieces and consequently a poor local approximation. We in this paper propose a simple approach that is robust across neighborhood widths in recovering faithful local explanations. In addition to a naive implementation of our approach which can still be accurate, we propose a novel adaptive neighborhood sampling scheme (ANS) that we formally show can be much more sample and query efficient. We then empirically evaluate our approach on real data where our explanations are significantly more sample and query efficient than the competitors, while also being faithful and stable across different widths.