Coherent Anti-Stokes Raman Scattering (CARS) has found critical applications across various fields, including high-speed chemical imaging, material science, and biomedical diagnostics. However, the inherent coherent nature of CARS poses challenges for quantitative chemical imaging due to the loss of spectral phase information. Accessing such information would enable faster chemical imaging speed through computational methods. Here, we develop a robust reference-less interferometric broadband pump/probe CARS to retrieve the vibrational spectral phase. We transfer the computational phase retrieval concept from quantitative spatial phase imaging to frequency-domain spectroscopy. We then unlock and demonstrate the concept of supervised compressive CARS microspectroscopy, enabling artifact-less high-speed quantitative chemical imaging.