Analysis of Different Types of Regret in Continuous Noisy Optimization

TL;DR

This paper investigates different types of regret measures in noisy optimization, demonstrating that some approximations like Approximate Simple Regret can misestimate convergence rates, while discussing the pros and cons of Robust Simple Regret.

Contribution

It provides a comparative analysis of Simple Regret approximations in noisy optimization, highlighting limitations of Approximate Simple Regret and evaluating Robust Simple Regret.

Findings

Approximate Simple Regret fails to estimate true convergence rates.

Robust Simple Regret has specific advantages and disadvantages.

Different regret measures impact the evaluation of noisy optimization algorithms.

Abstract

The performance measure of an algorithm is a crucial part of its analysis. The performance can be determined by the study on the convergence rate of the algorithm in question. It is necessary to study some (hopefully convergent) sequence that will measure how "good" is the approximated optimum compared to the real optimum. The concept of Regret is widely used in the bandit literature for assessing the performance of an algorithm. The same concept is also used in the framework of optimization algorithms, sometimes under other names or without a specific name. And the numerical evaluation of convergence rate of noisy algorithms often involves approximations of regrets. We discuss here two types of approximations of Simple Regret used in practice for the evaluation of algorithms for noisy optimization. We use specific algorithms of different nature and the noisy sphere function to show the following results. The approximation of Simple Regret, termed here Approximate Simple Regret, used in some optimization testbeds, fails to estimate the Simple Regret convergence rate. We also discuss a recent new approximation of Simple Regret, that we term Robust Simple Regret, and show its advantages and disadvantages.