Decomposable Probability-of-Success Metrics in Algorithmic Search

TL;DR

This paper introduces decomposable success metrics for algorithmic search, enabling broader applicability of impossibility results across different machine learning paradigms by expressing success as linear operations on probability distributions.

Contribution

It defines decomposable metrics and proves bounds on success, generalizing prior impossibility results in machine learning search problems.

Findings

Decomposable metrics can be expressed as linear operations on probability distributions.

Theorems provide bounds on success for these metrics.

Generalizes existing impossibility results.

Abstract

Previous studies have used a specific success metric within an algorithmic search framework to prove machine learning impossibility results. However, this specific success metric prevents us from applying these results on other forms of machine learning, e.g. transfer learning. We define decomposable metrics as a category of success metrics for search problems which can be expressed as a linear operation on a probability distribution to solve this issue. Using an arbitrary decomposable metric to measure the success of a search, we demonstrate theorems which bound success in various ways, generalizing several existing results in the literature.