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Adaptive Weighting in Knowledge Distillation: An Axiomatic Framework for Multi-Scale Teacher Ensemble Optimization

Aaron R. Flouro, Shawn P. Chadwick

TL;DR

The paper develops an operator-agnostic axiomatic framework for adaptive weighting in multi-teacher knowledge distillation across token, task, and context scales. It introduces scale-specific axioms, a product-structure unification to form a unified hierarchical weight operator, and proves existence and non-uniqueness of conforming weight families. It establishes operator-agnostic convergence with rate $O(1/t)$, contraction-based fixed-point analysis, and perturbation and gradient-stability results, along with a safety-constrained extension that formulates Pareto trade-offs and KKT optimality conditions. The framework decouples theoretical guarantees from concrete weighting formulas, enabling principled design and analysis of robust, safe distillation under heterogeneity and deployment shift.

Abstract

Knowledge distillation with multiple teachers is increasingly used to improve robustness, efficiency, and safety, yet existing approaches rely largely on heuristic or implementation-specific weighting schemes. This paper develops an operator-agnostic axiomatic framework for adaptive weighting in multi-teacher knowledge distillation across three complementary scales: token, task, and context. We formalize structural conditions under which adaptive weighting operators are well-defined, admit multiple non-equivalent implementations, and can be hierarchically composed via product-structure normalization. Within this framework, we establish existence and non-uniqueness of conforming operators, characterize convergence of gradient-based optimization under standard assumptions, analyze stability and perturbation robustness, and provide an abstract formulation of safety-constrained distillation. The results decouple theoretical guarantees from specific weighting formulas, enabling principled analysis of adaptive distillation methods under heterogeneity, distribution shift, and safety constraints.

Adaptive Weighting in Knowledge Distillation: An Axiomatic Framework for Multi-Scale Teacher Ensemble Optimization

TL;DR

The paper develops an operator-agnostic axiomatic framework for adaptive weighting in multi-teacher knowledge distillation across token, task, and context scales. It introduces scale-specific axioms, a product-structure unification to form a unified hierarchical weight operator, and proves existence and non-uniqueness of conforming weight families. It establishes operator-agnostic convergence with rate , contraction-based fixed-point analysis, and perturbation and gradient-stability results, along with a safety-constrained extension that formulates Pareto trade-offs and KKT optimality conditions. The framework decouples theoretical guarantees from concrete weighting formulas, enabling principled design and analysis of robust, safe distillation under heterogeneity and deployment shift.

Abstract

Knowledge distillation with multiple teachers is increasingly used to improve robustness, efficiency, and safety, yet existing approaches rely largely on heuristic or implementation-specific weighting schemes. This paper develops an operator-agnostic axiomatic framework for adaptive weighting in multi-teacher knowledge distillation across three complementary scales: token, task, and context. We formalize structural conditions under which adaptive weighting operators are well-defined, admit multiple non-equivalent implementations, and can be hierarchically composed via product-structure normalization. Within this framework, we establish existence and non-uniqueness of conforming operators, characterize convergence of gradient-based optimization under standard assumptions, analyze stability and perturbation robustness, and provide an abstract formulation of safety-constrained distillation. The results decouple theoretical guarantees from specific weighting formulas, enabling principled analysis of adaptive distillation methods under heterogeneity, distribution shift, and safety constraints.
Paper Structure (37 sections, 16 theorems, 24 equations, 1 figure, 1 table)

This paper contains 37 sections, 16 theorems, 24 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Limitations of Uniform Weighting
  3. From Limitations to Adaptive Reweighting
  4. Contributions
  5. Relation to Prior Work
  6. Organization
  7. Preliminaries and Axiomatic Setup
  8. Notation
  9. Standing Analytical Assumptions
  10. Axiomatic Framework for Multi-Scale Adaptive Weighting
  11. Token-Level Adaptive Weighting
  12. Task-Level Adaptive Weighting
  13. Context-Level Adaptive Weighting
  14. Hierarchical Unification: Product-Structure Composition
  15. Unified Weight Operator
  16. ...and 22 more sections

Key Result

Theorem 3.8

There exist non-trivial weight function families satisfying Axioms ax:tok-norm--ax:tok-safety.

Figures (1)

  • Figure 1: Product-structure composition of adaptive weight operators. Each scale (token, task, context) defines a conforming weight operator addressing one dimension of heterogeneity. The unified weight operator inherits axiom satisfaction from its components via multiplicative composition, preserving normalization and boundedness guarantees.

Theorems & Definitions (51)

  • Example 2.1: Divergence Without Bounded Weights
  • Remark 2.2: Role of Weight Bounds
  • Remark 3.6: Sufficient Conditions for Token Axioms
  • Definition 3.7: Token-Weighted Ensemble
  • Theorem 3.8: Existence of Conforming Token Weight Operators
  • proof : Proof (Sketch)
  • Theorem 3.9: Non-Uniqueness of Token Weight Operators
  • proof : Proof (Sketch)
  • Definition 3.15: Task-Weighted Ensemble
  • Theorem 3.16: Existence and Non-Uniqueness of Task Weight Operators
  • ...and 41 more