The beam alignment (BA) problem consists in accurately aligning the transmitter and receiver beams to establish a reliable communication link in wireless communication systems. Existing BA methods search the entire beam space to identify the optimal transmit-receive beam pair. This incurs a significant latency when the number of antennas is large. In this work, we develop a bandit-based fast BA algorithm to reduce BA latency for millimeter-wave (mmWave) communications. Our algorithm is named Two-Phase Heteroscedastic Track-and-Stop (2PHT\&S). We first formulate the BA problem as a pure exploration problem in multi-armed bandits in which the objective is to minimize the required number of time steps given a certain fixed confidence level. By taking advantage of the correlation structure among beams that the information from nearby beams is similar and the heteroscedastic property that the variance of the reward of an arm (beam) is related to its mean, the proposed algorithm groups all beams into several beam sets such that the optimal beam set is first selected and the optimal beam is identified in this set after that. Theoretical analysis and simulation results on synthetic and semi-practical channel data demonstrate the clear superiority of the proposed algorithm vis-à-vis other baseline competitors.