Predictive health assessment is of vital importance for smarter battery management to ensure optimal and safe operations and thus make the most use of battery life. This paper proposes a general framework for battery aging prognostics in order to provide the predictions of battery knee, lifetime, state of health degradation, and aging rate variations, as well as the assessment of battery health. Early information is used to predict knee slope and other life-related information via deep multi-task learning, where the convolutional-long-short-term memorybayesian neural network is proposed. The structure is also used for online state of health and degradation rate predictions for the detection of accelerating aging. The two probabilistic predicted boundaries identify the accelerating aging regions for battery health assessment. To avoid wrong and premature alarms, the empirical model is used for data preprocessing and the slope is predicted together with the state of health via multi-task learning. A cloud-edge framework is considered where fine-tuning is adopted for performance improvement during cycling. The proposed general framework is flexible for adjustment to different practical requirements and can be extrapolated to other batteries aged under different conditions. The results indicate that the early predictions are improved using the proposed method compared to multiple single feature-based benchmarks, and the algorithm integration is improved. The sequence prediction is reliable for different predicted lengths with root mean square errors of less than 1.41%, and the detection of accelerating aging can guide reliable predictive health management.