Extension of Minimax for Algorithmic Lower Bounds

TL;DR

This paper explores the application of Yao's Minimax Theorem in robust algorithm design, revealing that optimizing the ratio of expectations is equivalent to optimizing the expectation of ratios, impacting how lower bounds are derived.

Contribution

It extends Minimax analysis by showing the equivalence between optimizing the ratio of expectations and the expectation of ratios in algorithmic lower bounds.

Findings

The ratio of expectations can be used interchangeably with the expectation of ratios in Minimax analysis.

This equivalence simplifies the derivation of lower bounds in robust algorithms.

The results apply to online algorithms and other settings where performance ratios are critical.

Abstract

This paper considers the use of Yao's Minimax Theorem in robust algorithm design, e.g., for online algorithms, where the algorithm designer aims to minimize the ratio of the algorithm's performance to the optimal performance. When applying Minimax, the objective becomes the expectation of these ratios. The main result of the paper is that it is equally valid for the objective (after applying Minimax) to be the ratio of the expectations.