A Theoretical Framework for Stochastic Activity Prediction in Tensor Accelerator Wallace-Tree Multipliers

Abstract

Tensor accelerator multipliers burn dynamic power on every clock cycle, even when sparse operands require very little internal switching. No existing technique addresses this: zero-detection requires exactly-zero operands, structural power gating requires an idle multiplier, and offline weight selection cannot respond to runtime data. This paper introduces Stochastic Activity Prediction (SAP), which closes this gap by examining the Hamming weight of arriving operands before the multiplier executes, predicting low switching activity, and freezing the inputs when a deterministic Safety Controller independently confirms the reuse is correct. Mispredictions cause missed savings, never wrong answers. Three formal results underpin SAP: (i) a Spectral Contraction Lemma proving that Wallace-tree activity depends on operand bit density, not bit position, establishing Lipschitz constant $Lφ= 3/2$ and prediction error below $10^{-13}$ for a 256-cycle window; (ii) an Information Retention Theorem showing $η_I \ge 1 - O(\log n/n)$, so one bit per cycle captures nearly all predictive information about $O(n^2)$ internal nodes; and (iii) a Bernoulli Optimality Theorem proving the chosen encoding is shown to be optimal, within the family of calibrated one-bit encoders of Hamming-weight statistics considered. SAP addresses the specific layer of the tensor accelerator power stack that existing techniques do not cover.