Chasing Nested Convex Bodies Nearly Optimally
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The convex body chasing problem, introduced by Friedman and Linial, is a competitive analysis problem on any normed vector space. In convex body chasing, for each timestep t∈ N, a convex body K_t⊆ R^d is given as a request, and the player picks a point x_t∈ K_t. The player aims to ensure that the total distance ∑_t=0^T-1||x_t-x_t+1|| is within a bounded ratio of the smallest possible offline solution. In this work, we consider the nested version of the problem, in which the sequence (K_t) must be decreasing. For Euclidean spaces, we consider a memoryless algorithm which moves to the so-called Steiner point, and show that in a certain sense it is exactly optimal among memoryless algorithms. For general finite dimensional normed spaces, we combine the Steiner point and our recent previous algorithm to obtain a new algorithm which is nearly optimal for all ℓ^p_d spaces with p≥ 1, closing a polynomial gap.
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