The Famine of Forte: Few Search Problems Greatly Favor Your Algorithm

TL;DR

This paper demonstrates that only a small fraction of problems are favorable for any fixed algorithm, explaining the need for flexible models and new methods in machine learning, supported by bounds based on information theory.

Contribution

It provides theoretical bounds on the proportion of problems favorable to fixed algorithms and introduces a framework linking success probabilities to mutual information and search strategies.

Findings

Favorable problems for a fixed algorithm are strictly limited.

The success probability relates to mutual information between target and data.

Favorable search strategies are also rare and bounded.

Abstract

Casting machine learning as a type of search, we demonstrate that the proportion of problems that are favorable for a fixed algorithm is strictly bounded, such that no single algorithm can perform well over a large fraction of them. Our results explain why we must either continue to develop new learning methods year after year or move towards highly parameterized models that are both flexible and sensitive to their hyperparameters. We further give an upper bound on the expected performance for a search algorithm as a function of the mutual information between the target and the information resource (e.g., training dataset), proving the importance of certain types of dependence for machine learning. Lastly, we show that the expected per-query probability of success for an algorithm is mathematically equivalent to a single-query probability of success under a distribution (called a search strategy), and prove that the proportion of favorable strategies is also strictly bounded. Thus, whether one holds fixed the search algorithm and considers all possible problems or one fixes the search problem and looks at all possible search strategies, favorable matches are exceedingly rare. The forte (strength) of any algorithm is quantifiably restricted.