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The design of the buffer manager in database management systems (DBMSs) is influenced by the performance characteristics of volatile memory (i.e., DRAM) and non-volatile storage (e.g., SSD). The key design assumptions have been that the data must be migrated to DRAM for the DBMS to operate on it and that storage is orders of magnitude slower than DRAM. But the arrival of new non-volatile memory (NVM) technologies that are nearly as fast as DRAM invalidates these previous assumptions.Researchers have recently designed Hymem, a novel buffer manager for a three-tier storage hierarchy comprising of DRAM, NVM, and SSD. Hymem supports cache-line-grained loading and an NVM-aware data migration policy. While these optimizations improve its throughput, Hymem suffers from two limitations. First, it is a single-threaded buffer manager. Second, it is evaluated on an NVM emulation platform. These limitations constrain the utility of the insights obtained using Hymem. In this paper, we present Spitfire, a multi-threaded, three-tier buffer manager that is evaluated on Optane Persistent Memory Modules, an NVM technology that is now being shipped by Intel. We introduce a general framework for reasoning about data migration in a multi-tier storage hierarchy. We illustrate the limitations of the optimizations used in Hymem on Optane and then discuss how Spitfire circumvents them. We demonstrate that the data migration policy has to be tailored based on the characteristics of the devices and the workload. Given this, we present a machine learning technique for automatically adapting the policy for an arbitrary workload and storage hierarchy. Our experiments show that Spitfire works well across different workloads and storage hierarchies.