  Querying complex models for precise information (e.g. traffic models,
database systems, large ML models) often entails intense computations and
results in long response times. Thus, weaker models which give imprecise
results quickly can be advantageous, provided inaccuracies can be resolved
using few queries to a stronger model. In the fundamental problem of computing
a maximum-weight basis of a matroid, a well-known generalization of many
combinatorial optimization problems, algorithms have access to a clean oracle
to query matroid information. We additionally equip algorithms with a fast but
dirty oracle modelling an unknown, potentially different matroid. We design and
analyze practical algorithms which only use few clean queries w.r.t. the
quality of the dirty oracle, while maintaining robustness against arbitrarily
poor dirty matroids, approaching the performance of classic algorithms for the
given problem. Notably, we prove that our algorithms are, in many respects,
best-possible. Further, we outline extensions to other matroid oracle types,
non-free dirty oracles and other matroid problems.
