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The Role of Dimension in the Online Chasing Problem

Hristo Papazov

TL;DR

This work investigates how geometric and metric-dimensional properties govern online chasing problems. It introduces a universal embedding-based framework to transfer competitive-ratio bounds between chasing settings, enabling dimension-based lower bounds for nested CBC in arbitrary normed spaces from Euclidean results via Banach-Mazur distance. The authors demonstrate that doubling and Assouad dimensions do not constrain the hardness of ball chasing: for any $\\gamma>1$ there exists spaces with $\\text{dim}_{\\gamma}(X)=\\Theta_{\\gamma}(\\rho)$ where NBC is intractable, and there are spaces with fixed Assouad dimension where NBC is unbounded. They also provide a simple greedy upper bound for NBC and clarify the sharp dependence of CBC on dimension in Euclidean and $\\ell_p$ settings, yielding a lower bound of $\\Omega(d^{1/6})$ in general and up to $\\Omega(\,\

Abstract

Let $(X, d)$ be a metric space and $C \subseteq 2^X$ -- a collection of special objects. In the $(X,d,C)$-chasing problem, an online player receives a sequence of online requests $\{B_t\}_{t=1}^T \subseteq C$ and responds with a trajectory $\{x_t\}_{t=1}^T$ such that $x_t \in B_t$. This response incurs a movement cost $\sum_{t=1}^T d(x_t, x_{t-1})$, and the online player strives to minimize the competitive ratio -- the worst case ratio over all input sequences between the online movement cost and the optimal movement cost in hindsight. Under this setup, we call the $(X,d,C)$-chasing problem $\textit{chaseable}$ if there exists an online algorithm with finite competitive ratio. In the case of Convex Body Chasing (CBC) over real normed vector spaces, (Bubeck et al. 2019) proved the chaseability of the problem. Furthermore, in the vector space setting, the dimension of the ambient space appears to be the factor controlling the size of the competitive ratio. Indeed, recently, (Sellke 2020) provided a $d-$competitive online algorithm over arbitrary real normed vector spaces $(\mathbb{R}^d, ||\cdot||)$, and we will shortly present a general strategy for obtaining novel lower bounds of the form $Ω(d^c), \enspace c > 0$, for CBC in the same setting. In this paper, we also prove that the $\textit{doubling}$ and $\textit{Assouad}$ dimensions of a metric space exert no control on the hardness of ball chasing over the said metric space. More specifically, we show that for any large enough $ρ\in \mathbb{R}$, there exists a metric space $(X,d)$ of doubling dimension $Θ(ρ)$ and Assouad dimension $ρ$ such that no online selector can achieve a finite competitive ratio in the general ball chasing regime.

The Role of Dimension in the Online Chasing Problem

TL;DR

This work investigates how geometric and metric-dimensional properties govern online chasing problems. It introduces a universal embedding-based framework to transfer competitive-ratio bounds between chasing settings, enabling dimension-based lower bounds for nested CBC in arbitrary normed spaces from Euclidean results via Banach-Mazur distance. The authors demonstrate that doubling and Assouad dimensions do not constrain the hardness of ball chasing: for any there exists spaces with where NBC is intractable, and there are spaces with fixed Assouad dimension where NBC is unbounded. They also provide a simple greedy upper bound for NBC and clarify the sharp dependence of CBC on dimension in Euclidean and settings, yielding a lower bound of in general and up to $\\Omega(\,\

Abstract

Let be a metric space and -- a collection of special objects. In the -chasing problem, an online player receives a sequence of online requests and responds with a trajectory such that . This response incurs a movement cost , and the online player strives to minimize the competitive ratio -- the worst case ratio over all input sequences between the online movement cost and the optimal movement cost in hindsight. Under this setup, we call the -chasing problem if there exists an online algorithm with finite competitive ratio. In the case of Convex Body Chasing (CBC) over real normed vector spaces, (Bubeck et al. 2019) proved the chaseability of the problem. Furthermore, in the vector space setting, the dimension of the ambient space appears to be the factor controlling the size of the competitive ratio. Indeed, recently, (Sellke 2020) provided a competitive online algorithm over arbitrary real normed vector spaces , and we will shortly present a general strategy for obtaining novel lower bounds of the form , for CBC in the same setting. In this paper, we also prove that the and dimensions of a metric space exert no control on the hardness of ball chasing over the said metric space. More specifically, we show that for any large enough , there exists a metric space of doubling dimension and Assouad dimension such that no online selector can achieve a finite competitive ratio in the general ball chasing regime.
Paper Structure (15 sections, 18 theorems, 34 equations, 2 figures)

This paper contains 15 sections, 18 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction and Related Work
  2. Our Results.
  3. Competitive Ratio Bounds through Metric Embeddings
  4. Metric-Dimension Preliminaries
  5. Arbitrary Bad Competitive Ratio in Nested Ball Chasing
  6. Defining the Metric Space.
  7. Proof of Valid Metric.
  8. Nested Sequence of Balls.
  9. Bounding the Doubling Dimension.
  10. Bounding the Competitive Ratio.
  11. Greedy Upper Bound on $\hat{R}(X)$ for Nested Ball Chasing
  12. Infinite Competitive Ratio for Any Fixed Assouad Dimension
  13. Phase I.
  14. Phase II.
  15. Proofs of Metric-Dimension Lemmas

Key Result

Theorem 1.2

For every $d \in {\mathbb{N}}$ and $p \in [0, \infty]$, no online algorithm can achieve a competitive ratio better than ${\Omega} (\max(d^{1 - \frac{1}{p}}, \sqrt{d}))$ in the nested convex body chasing problem over $\ell_p^d$. Furthermore, no online algorithm can achieve a competitive ratio better

Figures (2)

  • Figure 1: $(X,d)$ with ${\texttt{dim}} _ {\gamma} (x) = \Theta_ {\gamma} (\rho)$ and $\hat{R}(X) \geq N$.
  • Figure 2: $(X,d)$ with ${\texttt{dim}} _ {\mathcal{A}} (x) = \rho$ and $\hat{R}(X) = \infty$.

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Lemma 3.1
  • ...and 24 more