Multi-objective multi-task learning aims to boost the performance of all tasks by leveraging their correlation and conflict appropriately. Nevertheless, in real practice, users may have preference for certain tasks, and other tasks simply serve as privileged or auxiliary tasks to assist the training of target tasks. The privileged tasks thus possess less or even no priority in the final task assessment by users. Motivated by this, we propose a privileged multiple descent algorithm to arbitrate the learning of target tasks and privileged tasks. Concretely, we introduce a privileged parameter so that the optimization direction does not necessarily follow the gradient from the privileged tasks, but concentrates more on the target tasks. Besides, we also encourage a priority parameter for the target tasks to control the potential distraction of optimization direction from the privileged tasks. In this way, the optimization direction can be more aggressively determined by weighting the gradients among target and privileged tasks, and thus highlight more the performance of target tasks under the unified multi-task learning context. Extensive experiments on synthetic and real-world datasets indicate that our method can achieve versatile Pareto solutions under varying preference for the target tasks.