TY - GEN

T1 - Chasing nested convex bodies nearly optimally

AU - Bubeck, Sébastien

AU - Klartag, Bo'az

AU - Lee, Yin Tat

AU - Li, Yuanzhi

AU - Sellke, Mark

N1 - Publisher Copyright:
Copyright © 2020 by SIAM

PY - 2020

Y1 - 2020

N2 - The convex body chasing problem, introduced by Friedman and Linial [FL93], is a competitive analysis problem on any normed vector space. In convex body chasing, for each timestep t ∈ N, a convex body Kt ⊆ Rd is given as a request, and the player picks a point xt ∈ Kt. The player aims to ensure that the total distance moved PTt=0−1 ||xt−xt+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 (Kt) must be decreasing. For Euclidean spaces, we consider a memoryless algorithm which moves to the so-called Steiner point, and show that in an appropriate sense it is exactly optimal among memoryless algorithms. For general finite dimensional normed spaces, we combine the Steiner point and our recent algorithm in [ABC+19] to obtain a new algorithm which is nearly optimal for all `pd spaces with p ≥ 1, closing a polynomial gap.

AB - The convex body chasing problem, introduced by Friedman and Linial [FL93], is a competitive analysis problem on any normed vector space. In convex body chasing, for each timestep t ∈ N, a convex body Kt ⊆ Rd is given as a request, and the player picks a point xt ∈ Kt. The player aims to ensure that the total distance moved PTt=0−1 ||xt−xt+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 (Kt) must be decreasing. For Euclidean spaces, we consider a memoryless algorithm which moves to the so-called Steiner point, and show that in an appropriate sense it is exactly optimal among memoryless algorithms. For general finite dimensional normed spaces, we combine the Steiner point and our recent algorithm in [ABC+19] to obtain a new algorithm which is nearly optimal for all `pd spaces with p ≥ 1, closing a polynomial gap.

UR - http://www.scopus.com/inward/record.url?scp=85084040491&partnerID=8YFLogxK

M3 - פרסום בספר כנס

AN - SCOPUS:85084040491

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1496

EP - 1508

BT - 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020

A2 - Chawla, Shuchi

PB - Association for Computing Machinery

Y2 - 5 January 2020 through 8 January 2020

ER -