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We address the problem of sequential prediction with expert advice in anon-stationary environment with long-term memory guarantees in the sense ofBousquet and Warmuth [4]. We give a linear-time algorithm that improves on thebest known regret bounds [26]. This algorithm incorporates a relative entropyprojection step. This projection is advantageous over previous weight-sharingapproaches in that weight updates may come with implicit costs as in forexample portfolio optimization. We give an algorithm to compute this projectionstep in linear time, which may be of independent interest.

Improved Regret Bounds for Tracking Experts with Memory