Recursive Meta-Distillation: An Axiomatic Framework for Iterative Knowledge Refinement

TL;DR

This paper introduces an axiomatic, operator-theoretic framework for recursive knowledge distillation, providing foundational insights into its convergence, stability, and theoretical properties without relying on specific algorithms.

Contribution

It formalizes recursive distillation as a sequence of probability operators, proving convergence and stability under mild assumptions, and offers a theoretical basis for understanding iterative knowledge refinement.

Findings

Recursive distillation induces contraction in KL divergence.

The framework guarantees geometric convergence to base teacher distributions.

It characterizes conditions for well-posed and stable recursive distillation.

Abstract

Recent work in probability-domain knowledge distillation has established axiomatic frameworks for temperature scaling, multi-teacher aggregation, and bias-variance trade-offs in single-stage settings. However, the mathematical behavior of recursive or multi-generation distillation remains poorly understood, with prior approaches relying primarily on empirical heuristics. In this work, we introduce an axiomatic and operator-theoretic framework for recursive meta-distillation, formalizing iterative knowledge distillation as a sequence of probability-distribution operators with explicit anchoring to base teachers. We define structural axioms for valid meta-teacher construction and prove the existence of non-trivial operator families satisfying these axioms without specifying particular algorithms or loss functions. Under mild realizability and convexity assumptions, we show that anchored recursive distillation induces contraction in KL divergence, yielding geometric convergence to base teacher distributions and a unique, globally attractive fixed point. The contribution is foundational rather than algorithmic: the framework characterizes when recursive distillation is mathematically well-posed and convergent rather than error-accumulating, independent of model architecture, optimization details, or specific operator instantiations. These results provide a theoretical basis for understanding stability, bias-variance behavior, and failure modes in iterative and multi-teacher distillation under capacity constraints.