The Optimal Choice of Hypothesis Is the Weakest, Not the Shortest (2301.12987v4)

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.

• The paper introduces weakness as a novel proxy for hypothesis generalization, challenging the traditional emphasis on minimal description length.
• Mathematical formalism and binary arithmetic experiments show that weaker hypotheses generalize 1.1 to 5 times better than shorter ones.
• The findings imply that optimizing AI systems for hypothesis weakness can lead to more adaptable and scalable learning across diverse tasks.

Evaluating the Weakness Proxy in Inductive Reasoning

Researchers have long debated the optimal strategies for hypothesis selection to maximize generalization capabilities in artificial intelligence systems. Historically, simplicity and minimization of description length have been synonymous with effective generalization, with proponents citing principles like Ockham's Razor and frameworks such as Minimum Description Length (MDL). In the paper "The Optimal Choice of Hypothesis Is the Weakest, Not the Shortest," the author Michael Timothy Bennett challenges this conventional wisdom and introduces an alternative criterion based on the concept of weakness.

Key Insights and Contributions

The central thesis of the paper argues that the minimization of description length is neither necessary nor sufficient for maximizing generalization. Instead, the paper proposes "weakness" as a superior proxy for predicting which hypotheses can generalize effectively. Weakness is defined as the cardinality of the extension of a statement, claiming that hypotheses which include larger extensions are more probable to generalize to unseen tasks.

The paper's claims are substantiated through mathematical formalism, employing the notion of enactive cognition. This approach considers tasks as constructs of situations, decisions, and models, elevating the focus from syntactic simplicity to semantic completeness regarding hypothesis confrontation with task expectations. Weakness, measured as the potential reach of a hypothesis across a variety of decision-making scenarios, outperforms traditional length-based proxies in controlled binary arithmetic experiments.

Numerical Results and Experimental Validation

Through experiments involving binary addition and multiplication tasks using a customized Python environment with PyTorch and SymPy, the paper demonstrates empirical backing for the theoretical claims. Models derived from the weakness proxy generalize at a rate between 1.1 to five times higher than those using minimum description length criteria.

Implications for Artificial Intelligence

The implications of these findings are profound for the fields of machine learning and artificial general intelligence (AGI). By redefining the criteria for proxy selection in hypothesis induction, AI systems can achieve more adaptable and scalable learning processes. Traditional compressive representations cause systems to underrepresent complexity, but by maximizing weakness, these systems could handle variance and incomplete information more effectively.

The application of the weakness proxy is particularly relevant to inference systems such as Deepmind's Apperception Engine, where hypothesis generation uses universally quantified expressions. Within such systems, the widespread applicability of task inference becomes more feasible, supporting further advances in understanding symbolic and enactive cognition in computational contexts.

Future Directions

The paper opens new lines of inquiry for applying the weakness maximization principle across diverse AI architectures, such as neural networks. The paper posits that optimizing neural architectures for weakness, rather than mere loss minimization, might reduce tendencies toward error and inconsistency, such as fabrication in LLMs.

Furthermore, this approach suggests that traditional vocabularies in computational learning systems could be reconsidered or tailored for specific tasks to optimize induction efficiency—a notion previously unexplored in many machine learning frameworks.

Broader adoption of the weakness proxy could reevaluate the baselines for what constitutes intelligent computational reasoning, allowing AI to engage more meaningfully and consistently with complex problem domains. Overall, Michael Timothy Bennett's paper provides a compelling case for reevaluating entrenched beliefs about simplicity and its role in AI generalization tasks, laying the groundwork for significant advancements in proxy optimization and intelligence estimation.