Merging mobile edge computing with the dense deployment of small cell base stations promises enormous benefits such as a real proximity, ultra-low latency access to cloud functionalities. However, the envisioned integration creates many new challenges and one of the most significant is mobility management, which is becoming a key bottleneck to the overall system performance. Simply applying existing solutions leads to poor performance due to the highly overlapped coverage areas of multiple base stations in the proximity of the user and the co-provisioning of radio access and computing services. In this paper, we develop a novel user-centric mobility management scheme, leveraging Lyapunov optimization and multi-armed bandits theories, in order to maximize the edge computation performance for the user while keeping the user's communication energy consumption below a constraint. The proposed scheme effectively handles the uncertainties present at multiple levels in the system and provides both short-term and long-term performance guarantee. Simulation results show that our proposed scheme can significantly improve the computation performance (compared to state of the art) while satisfying the communication energy constraint.