The Optimal Choice of Hypothesis Is the Weakest, Not the Shortest

Abstract

If $A$ and $B$ are sets such that $A\subset B$, generalisation may be understood as the inference from $A$ of a hypothesis sufficient to construct $B$. One might infer any number of hypotheses from $A$, yet only some of those may generalise to $B$. How can one know which are likely to generalise? One strategy is to choose the shortest, equating the ability to compress information with the ability to generalise (a proxy for intelligence). We examine this in the context of a mathematical formalism of enactive cognition. We show that compression is neither necessary nor sufficient to maximise performance (measured in terms of the probability of a hypothesis generalising). We formulate a proxy unrelated to length or simplicity, called weakness. We show that if tasks are uniformly distributed, then there is no choice of proxy that performs at least as well as weakness maximisation in all tasks while performing strictly better in at least one. In experiments comparing maximum weakness and minimum description length in the context of binary arithmetic, the former generalised at between $1.1$ and $5$ times the rate of the latter. We argue this demonstrates that weakness is a far better proxy, and explains why Deepmind's Apperception Engine is able to generalise effectively.

arXiv commentPublished at the 16th Conference on Artificial General Intelligence, Stockholm, 2023

References in corpus (6)

• PyTorch: An Imperative Style, High-Performance Deep Learning Library
• The Symbol Grounding Problem
• Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets
• Neural Networks and the Chomsky Hierarchy
• Compression, The Fermi Paradox and Artificial Super-Intelligence
• Symbol Emergence and The Solutions to Any Task

Cited by in corpus (2)

• Is Complexity an Illusion?
• Computational Dualism and Objective Superintelligence