[Paper Review] Harmonizing Program Induction with Rate-Distortion Theory

Hanqi Zhou, Dávid Nagy+1 more|arXiv (Cornell University)|May 8, 2024

Analog and Mixed-Signal Circuit Design4 citations

TL;DR

This paper integrates Rate-Distortion Theory (RDT) with program induction by introducing a three-way trade-off among rate (program description length), distortion (reconstruction error), and search cost (computational budget). Using melody learning simulations, it shows that curriculum-designed program libraries—optimized via partial information decomposition to maximize synergy—enable more efficient generalization and resource-efficient learning than random ordering.

ABSTRACT

Many aspects of human learning have been proposed as a process of constructing mental programs: from acquiring symbolic number representations to intuitive theories about the world. In parallel, there is a long-tradition of using information processing to model human cognition through Rate Distortion Theory (RDT). Yet, it is still poorly understood how to apply RDT when mental representations take the form of programs. In this work, we adapt RDT by proposing a three way trade-off among rate (description length), distortion (error), and computational costs (search budget). We use simulations on a melody task to study the implications of this trade-off, and show that constructing a shared program library across tasks provides global benefits. However, this comes at the cost of sensitivity to curricula, which is also characteristic of human learners. Finally, we use methods from partial information decomposition to generate training curricula that induce more effective libraries and better generalization.

Motivation & Objective

• To extend Rate-Distortion Theory (RDT) to mental program representations, which are inherently structured and path-dependent.
• To investigate how cognitive resource constraints—specifically, description length and search budget—affect program induction and generalization.
• To explore whether curriculum design guided by information-theoretic synergy can improve the efficiency and effectiveness of program library development.
• To model human-like learning behaviors such as sensitivity to learning order and the emergence of shared, reusable program components.

Proposed method

• Proposes a three-way trade-off in program induction: rate (R_l, description length), distortion (d(X, X̂), reconstruction error), and search cost (R_s, computational budget).
• Uses an encoder-decoder framework where melodies (piano rolls) are encoded into latent programs π via a program library 𝒮, with constraints on R_l and R_s.
• Employs Adaptor Grammars (AGs) to represent programs as hierarchical trees with routers, enabling compositional generalization and subprogram reuse.
• Applies Partial Information Decomposition (PID) to quantify synergy S(ℒ;X) among programs in a library, defined as the shared information beyond individual and unique contributions.
• Designs a curriculum strategy that selects the next melody X* to maximize future library synergy: X* = argmax_X S(ℒ;X), where ℒ is updated based on prior library and new input.
• Uses a term deletion process to fine-tune rate control, enabling precise manipulation of program description length while maintaining reconstruction fidelity.

Experimental results

Research questions

• RQ1 How does incorporating search cost as a constraint in RDT affect program induction and representation learning?
• RQ2 Can a shared program library across tasks improve generalization efficiency in program induction under resource constraints?
• RQ3 Does curriculum design based on synergy (via Partial Information Decomposition) lead to more effective and generalizable program libraries compared to random ordering?
• RQ4 How do path-dependent learning dynamics—shaped by curriculum order—affect the emergence of reusable program components?

Key findings

• The proposed three-way trade-off among rate, distortion, and search cost enables more efficient and cognitively plausible program induction than traditional RDT alone.
• Constructing a shared program library across tasks leads to better generalization performance, as the library captures reusable structural patterns across melodies.
• Curricula designed to maximize synergy (S(ℒ;X)) significantly improve generalization compared to random ordering, with a statistically significant t-test result: t(999) = 6.219, p < .001.
• The synergy-based curriculum fosters the development of more effective program libraries, as evidenced by enhanced representational capacity and reduced reconstruction error.
• The model demonstrates sensitivity to curriculum order, mirroring human learning patterns where learning sequence influences the efficiency and structure of mental program development.
• The term deletion process enables fine-grained control over rate, though it may overestimate distortion under certain constraints, suggesting a need for refinement in cognitive plausibility.