For convex losses in nested evolving feasible sets, a lazy algorithm balances O(T^{1-β}) regret with O(T^β) movement for any β; for strongly convex or sharp losses, Frugal achieves zero regret with O(log T) movement, shown optimal by matching lower bound.
On convex body chasing.Discrete & Computational Geometry, 9(3):293–321
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Convex Optimization with Nested Evolving Feasible Sets
For convex losses in nested evolving feasible sets, a lazy algorithm balances O(T^{1-β}) regret with O(T^β) movement for any β; for strongly convex or sharp losses, Frugal achieves zero regret with O(log T) movement, shown optimal by matching lower bound.