Annual low dose computed tomography (CT) lung screening is currently advised for individuals at high risk of lung cancer (e.g., heavy smokers between 55 and 80 years old). The recommended screening practice significantly reduces all-cause mortality, but the vast majority of screening results are negative for cancer. If patients at very low risk could be identified based on individualized, image-based biomarkers, the health care resources could be more efficiently allocated to higher risk patients and reduce overall exposure to ionizing radiation. In this work, we propose a multi-task (diagnosis and prognosis) deep convolutional neural network to improve the diagnostic accuracy over a baseline model while simultaneously estimating a personalized cancer-free progression time (CFPT). A novel Censored Regression Loss (CRL) is proposed to perform weakly supervised regression so that even single negative screening scans can provide small incremental value. Herein, we study 2287 scans from 1433 de-identified patients from the Vanderbilt Lung Screening Program (VLSP) and Molecular Characterization Laboratories (MCL) cohorts. Using five-fold cross-validation, we train a 3D attention-based network under two scenarios: (1) single-task learning with only classification, and (2) multi-task learning with both classification and regression. The single-task learning leads to a higher AUC compared with the Kaggle challenge winner pre-trained model (0.878 v. 0.856), and multi-task learning significantly improves the single-task one (AUC 0.895, p<0.01, McNemar test). In summary, the image-based predicted CFPT can be used in follow-up year lung cancer prediction and data assessment.