  Randomized measurement protocols such as classical shadows represent powerful
resources for quantum technologies, with applications ranging from quantum
state characterization and process tomography to machine learning and error
mitigation. Recently, the notion of measurement dual frames, in which classical
shadows are generalized to dual operators of POVM effects, resurfaced in the
literature. This brought attention to additional degrees of freedom in the
post-processing stage of randomized measurements that are often neglected by
established techniques. In this work, we leverage dual frames to construct
improved observable estimators from informationally complete measurement
samples. We introduce novel classes of parametrized frame superoperators and
optimization-free dual frames based on empirical frequencies, which offer
advantages over their canonical counterparts while retaining computational
efficiency. Remarkably, this comes at almost no quantum or classical cost, thus
rendering dual frame optimization a valuable addition to the randomized
measurement toolbox.
