Neural architecture search (NAS) has attracted a lot of attention in recent years. Researchers aim to develop the most efficient algorithms to automate the time-consuming task of architecture design. As a result, many complex methods are being developed in this area. Recent works, however, proved that a simple Local Search (LS) method is competitive with state-of-the-art techniques. On top of its easy implementation, LS has many advantages in the world of NAS. It can naturally exploit strategies that accelerate global search time like network morphism and weight inheritance. In the present work, we aim at making LS even more efficient by optimizing the exploration of the neighborhood using a weighted sampling. Our method, called LS-Weights, is easy to implement and keeps all advantages of local search while being much faster. Results on NASBench-201 show that it significantly improves the speed of local search without degrading its accuracy performance.