Multi-contrast MRI provides a comprehensive picture of tissue microstructure, but the high dimensionality of the parameter space increases scan time. In this work, we present a data-driven approach to multi-contrast MRI experiment design using concrete autoencoders. Concrete autoencoders simultaneously perform measurement subset-selection and learn a prediction of the full set of measurements. This approach was evaluated on two multi-contrast databases encoding diffusion, relaxation, and susceptibility. The results showed similar patterns of measurement-subset selection and mean-squared errors across different training sets. The increasing availability of public multi-contrast MRI databases can further push data-driven approaches in providing recommendations for experiment design.