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arXiv · 2609.31641

Product-Aware Deterministic Rounding for Quantized Matrix Multiplication

Piyush Sao·Narasinga Miniskar·Pedro Valero-Lara·Keita Teranishi·Sudip Seal
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Abstract

Scalar rounding decisions interact through matrix multiplication. We study deterministic product-aware rounding after scales, clipping bounds, and grids are fixed, with each active scalar choosing between adjacent levels. For dynamic activation rounding, null-space reduction preserves the relaxed product while leaving at most $r$ fractional decisions, where $r$ is the rank of the active gap-weighted weight block. Conditional-expectation completion gives a deterministic polynomial-time algorithm with squared product error at most $ \mathrm{OPT}_{\mathrm{dyn}}+rν_{\max}^2/4$, where $\mathrm{OPT}_{\mathrm{dyn}}$ is the best admissible error and $ν_{\max}$ is the largest row norm of that block. For reusable static weights, the exact expected product-loss metric is the uncentered input second moment with fixed output bias; free bias recalibration yields the centered covariance. Exact optimization is NP-hard even at rank one. In balanced blocks with $K=1024$ and $r=16$, conditional- expectation completion attains a dither-normalized median error of $0.010$, compared with $0.899$ for round-to-nearest. Clipping-aware initialization reduces median normalized error by a factor of $43.4$ at ten-percent clipping. Held-out Digits experiments show that retaining the input mean or correcting the output bias improves median product error over round-to-nearest in all four tested bit-width and calibration-size settings.

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Piyush Sao, Narasinga Miniskar, Pedro Valero-Lara, Keita Teranishi, Sudip Seal. 2026-09-11. Product-Aware Deterministic Rounding for Quantized Matrix Multiplication. https://arxiv.org/abs/2609.31641

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