Chasing Convex Bodies Optimally

@inproceedings{Sellke2020ChasingCB,
  title={Chasing Convex Bodies Optimally},
  author={Mark Sellke},
  booktitle={SODA},
  year={2020}
}
  • Mark Sellke
  • Published in SODA 2020
  • Computer Science, Mathematics
In the chasing convex bodies problem, an online player receives a request sequence of $N$ convex sets $K_1,\dots, K_N$ contained in a normed space $\mathbb R^d$. The player starts at $x_0\in \mathbb R^d$, and after observing each $K_n$ picks a new point $x_n\in K_n$. At each step the player pays a movement cost of $||x_n-x_{n-1}||$. The player aims to maintain a constant competitive ratio against the minimum cost possible in hindsight, i.e. knowing all requests in advance. The existence of a… Expand
Nested Convex Bodies are Chaseable
TLDR
This work gives a f ( d )-competitive algorithm for chasing nested convex bodies in R d, which retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial’s conjecture. Expand
Chasing Convex Bodies with Linear Competitive Ratio
TLDR
An algorithm is given that is O(\min(d, \sqrt{d \log T}))-competitive for any sequence of length $T$. Expand
Online Multiserver Convex Chasing and Optimization
TLDR
The problem of k-chasing of convex functions is introduced, a simultaneous generalization of both the famous k-server problem in $R^d$ and of the problem of chasing convex bodies and functions, for which it is shown that competitive online algorithms exist, and moreover with dimension-free competitive ratio. Expand
Dimension-Free Bounds on Chasing Convex Functions
TLDR
The problem of chasing convex functions, where functions arrive over time, is considered, and an algorithm is given that achieves an $O(\sqrt \kappa)$-competitiveness, when the functions are supported on $k$-dimensional affine subspaces. Expand
Fully-Dynamic Submodular Cover with Bounded Recourse
  • Anupam Gupta, Roie Levin
  • Computer Science, Mathematics
  • 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
  • 2020
TLDR
This work simultaneously simplifies and unifies previous results, as well as generalizes to a significantly larger class of covering problems, which includes set-coverage functions, and extensively uses the idea of Mutual Coverage, which generalizes the classic notion of mutual information. Expand
Lipschitz Selectors may not Yield Competitive Algorithms for Convex Body Chasing
TLDR
The current best algorithms for convex body chasing problem in online algorithms use the notion of the Steiner point of a convex set, and it is shown that being Lipschitz with respect to an Lp-type analog to the Hausdorff distance is sufficient to guarantee competitiveness if and only if p = 1. Expand
A PTAS for Euclidean TSP with Hyperplane Neighborhoods
TLDR
A novel and general sparsification technique is developed to transform an arbitrary convex polytope into one with a constant number of vertices and, in turn, into one of bounded complexity in the above sense. Expand
The Online Min-Sum Set Cover Problem
TLDR
A memoryless online algorithm, called Move-All-Equally, is proposed, which is inspired by the Double Coverage algorithm for the k-server problem and obtained (almost) tight bounds on the competitive ratio of deterministic online algorithms for MSSC. Expand
Scale-Free Allocation, Amortized Convexity, and Myopic Weighted Paging
TLDR
A natural myopic model for weighted paging in which an algorithm has access to the relative ordering of all pages with respect to the time of their next arrival is considered, which provides an $\ell$-competitive deterministic and an $O(\log \ell)-competitive randomized algorithm, where $ell$ is the number of distinct weight classes. Expand
Online Optimization with Predictions and Non-convex Losses
TLDR
This work gives two general sufficient conditions that specify a relationship between the hitting and movement costs which guarantees that a new algorithm, Synchronized Fixed Horizon Control (SFHC), achieves a 1+O(1/w) competitive ratio, where w is the number of predictions available to the learner. Expand
...
1
2
3
...

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. Expand
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. Expand
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. Expand
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). Expand
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. Expand
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. Expand
A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs
TLDR
This work considers a natural online optimization problem set on the real line, where at each integer time, a convex function arrives online and the online algorithm picks a new location, and gives a 2-competitive algorithm for this problem. Expand
Introduction to Online Convex Optimization
  • Elad Hazan
  • Computer Science, Mathematics
  • Found. Trends Optim.
  • 2016
TLDR
This monograph portrays optimization as a process, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. Expand
An Elementary Introduction to Modern Convex Geometry
Preface 1 Lecture 1. Basic Notions 2 Lecture 2. Spherical Sections of the Cube 8 Lecture 3. Fritz John’s Theorem 13 Lecture 4. Volume Ratios and Spherical Sections of the Octahedron 19 Lecture 5. TheExpand
Flavors of Geometry
This book collects accessible lectures on four geometrically flavored fields of mathematics that have experienced great development in recent years: hyperbolic geometry (taught by James Cannon),Expand
...
1
2
3
...