In the last-level cache, large amounts of blocks have reuse distances greater than the available cache capacity. Cache performance and efficiency can be improved if some subset of these distant reuse blocks can reside in the cache longer. The bypass technique is an effective and attractive solution that prevents the insertion of harmful blocks.Our analysis shows that bypass can contribute significant performance improvement, and the optimal bypass can achieve similar performance compared to OPT+B, which is the theoretical optimal replacement policy. Thus, we propose a bypass technique called Optimal Bypass Monitor (OBM), which makes bypass decisions by learning and predicting the behavior of the optimal bypass. OBM keeps a short global track of the incoming-victim block pairs. By detecting the first reuse block in each pair, the behavior of the optimal bypass on the track can be asserted to guide the bypass choice.Any existing replacement policy can be extended with OBM while requiring negligible design modification. Our experimental results show that using less than 1.5KB extra memory, OBM with the NRU replacement policy outperforms LRU by 9.7% and 8.9% for single-thread and multiprogrammed workloads respectively. Compared with other state-of-the-art proposals such as DRRIP and SDBP, it achieves superior performance with less storage overhead.