  We construct almost conserved local operators, that possess a minimal
commutator with the Hamiltonian of the system, near the many-body localization
transition of a one-dimensional disordered spin chain. We collect statistics of
these slow operators for different support sizes and disorder strengths, both
using exact diagonalization and tensor networks. Our results show that the
scaling of the average of the smallest commutators with the support size is
sensitive to Griffiths effects in the thermal phase and the onset of many-body
localization. Furthermore, we demonstrate that the probability distributions of
the commutators can be analyzed using extreme value theory and that their tails
reveal the difference between diffusive and sub-diffusive dynamics in the
thermal phase.
