Randomized online computation with high probability guarantees

TL;DR

This paper establishes that for a broad class of randomized online minimization problems, algorithms with constant expected competitive ratios can be transformed into algorithms that perform well with high probability, under certain conditions.

Contribution

It introduces a framework linking expected competitive ratios to high probability guarantees for online algorithms, including tightness and applicability to various problems.

Findings

High probability algorithms can match expected competitive ratios with small error

The assumptions for the results are shown to be tight

Applicable to well-studied problems like paging, k-server, and metrical task systems

Abstract

We study the relationship between the competitive ratio and the tail distribution of randomized online minimization problems. To this end, we define a broad class of online problems that includes some of the well-studied problems like paging, k-server and metrical task systems on finite metrics, and show that for these problems it is possible to obtain, given an algorithm with constant expected competitive ratio, another algorithm that achieves the same solution quality up to an arbitrarily small constant error a with high probability; the "high probability" statement is in terms of the optimal cost. Furthermore, we show that our assumptions are tight in the sense that removing any of them allows for a counterexample to the theorem. In addition, there are examples of other problems not covered by our definition, where similar high probability results can be obtained.