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In-line lensless digital holography has great potential in multiple applications; however, reconstructing high-quality images from a single recorded hologram is challenging due to the loss of phase information. Typical reconstruction methods are based on solving a regularized inverse problem and work well under suitable image priors, but they are extremely sensitive to mismatches between the forward model and the actual imaging system. This paper aims to improve the robustness of such algorithms by introducing the adaptive sparse reconstruction method, ASR, which learns a properly constrained point spread function (PSF) directly from data, as opposed to solely relying on physics-based approximations of it. ASR jointly performs holographic reconstruction, PSF estimation, and phase retrieval in an unsupervised way by maximizing the sparsity of the reconstructed images. Like traditional methods, ASR uses the image formation model along with a sparsity prior, which, unlike recent deep learning approaches, allows for unsupervised reconstruction with as little as one sample. Experimental results in synthetic and real data show the advantages of ASR over traditional reconstruction methods, especially in cases where the theoretical PSF does not match that of the actual system.