Chasing Convex Bodies with Linear Competitive Ratio

@inproceedings{Argue2020ChasingCB,
  title={Chasing Convex Bodies with Linear Competitive Ratio},
  author={C. J. Argue and Anupam Gupta and Guru Guruganesh and Ziye Tang},
  booktitle={SODA},
  year={2020}
}
We study the problem of chasing convex bodies online: given a sequence of convex bodies $K_t\subseteq \mathbb{R}^d$ the algorithm must respond with points $x_t\in K_t$ in an online fashion (i.e., $x_t$ is chosen before $K_{t+1}$ is revealed). The objective is to minimize the sum of distances between successive points in this sequence. Bubeck et al. (STOC 2019) gave a $2^{O(d)}$-competitive algorithm for this problem. We give an algorithm that is $O(\min(d, \sqrt{d \log T}))$-competitive for any… 

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References

SHOWING 1-10 OF 22 REFERENCES
Chasing Nested Convex Bodies Nearly Optimally
TLDR
This work considers the nested version of the convex body chasing problem, in which the sequence $(K_t)$ must be decreasing, and considers a memoryless algorithm which moves to the so-called Steiner point, and shows that in a certain sense it is exactly optimal among memoryless algorithms.
Chasing Convex Bodies Optimally
TLDR
The functional Steiner point of a convex function is defined and applied to the work function to obtain the algorithm achieving competitive ratio d for arbitrary normed spaces, which is exactly tight for $\ell^{\infty}$.
A Nearly-Linear Bound for Chasing Nested Convex Bodies
TLDR
A different strategy which is O(dlog d)-competitive algorithm for this nested convex body chasing problem, which works for any norm and is almost tight, given an Ω(d) lower bound for the l∞ norm.
Competitively chasing convex bodies
TLDR
This paper proves that the family of convex sets in Euclidean space is chaseable, which was conjectured in 1991 by Linial and Friedman to be chaseable.
Nested Convex Bodies are Chaseable
TLDR
The nested setting is closely related to extending the online LP framework of Buchbinder and Naor to arbitrary linear constraints and retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture.
Chasing Convex Bodies and Functions
TLDR
The first O(1)-competitive algorithm for Online Convex Optimization in two dimensions is obtained, and the first O-competitiveness analysis for chasing linear subspaces is given.
On convex body chasing
TLDR
This work provides a strategy for the player which is competitive, i.e., for any sequenceFi the cost to the player is within a constant (multiplicative) factor of the “off-line” cost (i.e, the least possible cost when allFi are known in advance).
Beyond Online Balanced Descent: An Optimal Algorithm for Smoothed Online Optimization
TLDR
A new lower bound is proved on the competitive ratio of any online algorithm in the setting where the costs are $m-strongly convex and the movement costs are the squared $\ell_2$ norm, showing that no algorithm can achieve a competitive ratio that is $o(m^{-1/2})$ as $m$ tends to zero.
Smoothed Online Convex Optimization in High Dimensions via Online Balanced Descent
TLDR
OBD is the first algorithm to achieve a dimension-free competitive ratio, 3 + O(1/\alpha)$, for locally polyhedral costs, where $\alpha$ measures the "steepness" of the costs.
Centres of Convex Sets inLpMetrics
It is shown that for each convex bodyA?Rnthere exists a naturally defined family GA?C(Sn?1) such that for everyg?GA, and every convex functionf:R?Rthe mappingy??Sn?1f(g(x)??y, x?)d?(x) has a
...
1
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3
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