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Applications in the cloud are vulnerable to several attack scenarios. In one possibility, an untrusted cloud operator can examine addresses on the memory bus and use this information leak to violate privacy guarantees, even if data is encrypted. The Oblivious RAM (ORAM) construct was introduced to eliminate such information leak and these frameworks have seen many innovations in recent years. In spite of these innovations, the overhead associated with ORAM is very significant. This paper takes a step forward in reducing ORAM memory bandwidth overheads. We make the case that, similar to a cache hierarchy, a lightweight ORAM that fronts the full-fledged ORAM provides a boost in efficiency. The lightweight ORAM has a smaller capacity and smaller depth, and it can relax some of the many constraints imposed on the full-fledged ORAM. This yields a 2-level hierarchy with a relaxed ORAM and a full ORAM. The relaxed ORAM adopts design parameters that are optimized for efficiency and not capacity. We introduce a novel metadata management technique to further reduce the bandwidth for relaxed ORAM access. Relaxed ORAM accesses preserve the indistinguishability property and are equipped with an integrity verification system. Finally, to eliminate information leakage through LLC and relaxed ORAM hit rates, we introduce a deterministic memory scheduling policy. On a suite of memory-intensive applications, we show that the best Relaxed Hierarchical ORAM (ρ) model yields a performance improvement of 50%, relative to a Freecursive ORAM baseline.