Chasing Convex Bodies Optimally (1905.11968v3)

Abstract: 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 finite competitive ratio for convex body chasing was first conjectured in 1991 by Friedman and Linial. This conjecture was recently resolved with an exponential $2^{O(d)}$ upper bound on the competitive ratio. We give an improved algorithm achieving competitive ratio $d$ in any normed space, which is exactly tight for $\ell^{\infty}$. In Euclidean space, our algorithm also achieves competitive ratio $O(\sqrt{d\log N})$, nearly matching a $\sqrt{d}$ lower bound when $N$ is subexponential in $d$. The approach extends our prior work for nested convex bodies, which is based on the classical Steiner point of a convex body. We define the functional Steiner point of a convex function and apply it to the associated work function.

• The paper presents an improved competitive ratio algorithm achieving a factor of d in general normed spaces and O(√(d log N)) in Euclidean spaces.
• The approach leverages the classical Steiner point concept to design a functional strategy that efficiently determines center points within convex bodies.
• The advancements bridge theoretical conjectures with practical implementations, paving the way for further research in online convex optimization.

An Analysis of Convex Body Chasing and Competitive Algorithms

This paper addresses the challenging problem of chasing convex bodies in the context of online algorithms, particularly focusing on minimizing the cost incurred while moving through a sequence of convex sets in a normed space. This problem, initially conjectured to have a finite competitive ratio for any dimension $d$, has witnessed significant advancements, with this work offering new insights and improved algorithmic solutions.

Overview of the Problem

The primary objective in the convex body chasing problem is to devise an online strategy that dictates the movement of a point to minimize total movement cost. This movement happens across a sequence of convex sets, $K_1, K_2, \dots, K_N$, in a normed space $X$. The key challenge arises from the lack of prior knowledge about future requests, which distinguishes it from offline solutions that could potentially organize optimal paths with complete future information.

Key Contributions

The paper offers significant contributions to the convex body chasing problem by suggesting an algorithm that remarkably improves the competitive ratio, which is a paramount criterion for evaluating the performance of online algorithms. Two main competitive ratios are highlighted:

1. Competitive Ratio in General Normed Spaces: The paper presents an improved algorithm achieving a competitive ratio of $d$, which is proven to be tight for $\ell^{\infty}$ spaces. This result provides a tight bound against which other algorithms' performance in such spaces can be measured.
2. Competitive Ratio in Euclidean Spaces: It further explores the context of Euclidean spaces, demonstrating that the algorithm can achieve an $O(\sqrt{d\log N})$ competitive ratio. This is particularly relevant because it aligns closely with pre-established lower bounds under certain conditions.

Theoretical Foundations and Tools

The algorithm builds upon the classical notion of the Steiner point, a well-established concept in mathematics used for selecting a point within a convex body. The Steiner point is employed here to effectively determine a center point within convex functions, extending the methodology previously applied to nested convex bodies.

The innovative application of the functional Steiner point, defined for convex functions, aligns convex function chasing with the broader context of metrical task systems. The introduction of the functional Steiner point and work function encapsulates the idea that strategies based on these constructs allow maintaining a bounded competitive ratio.

Implications and Future Directions

By refining the competitive ratio in higher dimensional spaces and for Euclidean contexts, this paper propels the field forward significantly. It bridges the gap between theoretical conjectures and practical algorithms that effectively tackle high-dimensional convex shape requests.

Future work could explore different normed space configurations beyond $\ell^{\infty}$ and Euclidean spaces, potentially exploring further applications in practical data-driven environments, such as real-time decision-making scenarios where the convex body chasing model can serve as a fundamental abstraction.

Conclusion

This paper provides a critical advancement in solving the convex body chasing problem, closing previously unresolved gaps by presenting an optimal competitive ratio algorithm applicable across various dimensional and normed spaces. The introduction, analysis, and implications of using the functional Steiner point solidify the paper's standing as a notable contribution to online algorithms and computational geometry.

These insights not only enhance the understanding of convex body chasing but also pave the way for further algorithmic developments in related fields, establishing foundational principles for future exploration and innovation in competitive online decision-making frameworks.