Sparse Knowledge Distillation: A Mathematical Framework for Probability-Domain Temperature Scaling and Multi-Stage Compression

• URL: http://arxiv.org/abs/2601.03195v1
• Date: Tue, 06 Jan 2026 17:17:24 GMT
• Title: Sparse Knowledge Distillation: A Mathematical Framework for Probability-Domain Temperature Scaling and Multi-Stage Compression
• Authors: Aaron R. Flouro, Shawn P. Chadwick,
• Abstract summary: We develop a unified theoretical framework for sparse knowledge distillation based on probability-domain softening operators.<n>We introduce an axiomatic definition of probability-domain softening operators based on ranking preservation, continuity, entropy monotonicity, identity, and boundary behavior.<n>Results provide theoretical grounding for black-box teacher distillation, partial-access settings such as top-$k$ truncation and text-only outputs, and privacy-equivalent model compression.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a unified theoretical framework for sparse knowledge distillation based on probability-domain softening operators. While the equivalence $p^{1/T} \propto \mathrm{softmax}(z/T)$ is well known, our contribution is an operator-level analytical framework built on this foundation rather than the equivalence itself. The framework comprises four core components: (i) operator-agnostic bias--variance decompositions that characterize when sparse students outperform dense teachers, (ii) a homotopy path formalization of multi-stage pruning in function space explaining why iterative compression succeeds where one-shot pruning fails, (iii) convergence guarantees establishing $O(1/n)$ rates for $n$-stage distillation with explicit parameter dependence, and (iv) equivalence class characterizations identifying distinct probability-domain operators that yield identical student models under capacity constraints. We introduce an axiomatic definition of probability-domain softening operators based on ranking preservation, continuity, entropy monotonicity, identity, and boundary behavior, and show that multiple non-equivalent operator families satisfy these axioms. All learning-theoretic guarantees are shown to hold uniformly across this operator class, independent of implementation details. These results provide theoretical grounding for black-box teacher distillation, partial-access settings such as top-$k$ truncation and text-only outputs, and privacy-preserving model compression.

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