The Optimal Choice of Hypothesis Is the Weakest, Not the Shortest
This addresses a foundational issue in machine learning and AI for researchers and practitioners by offering a novel criterion for hypothesis selection, though it is incremental in refining generalization theory.
The paper tackles the problem of selecting hypotheses that generalize well, challenging the idea that compression (shortest hypothesis) is optimal, and proposes 'weakness' as a better proxy, showing in experiments that it generalizes at 1.1 to 5 times the rate of minimum description length in binary arithmetic tasks.
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.