Product-Aware Deterministic Rounding for Quantized Matrix Multiplication

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

cs.LGSep 11, 2026

Product-Aware Deterministic Rounding for Quantized Matrix Multiplication

Authors: Piyush Sao, Narasinga Miniskar, Pedro Valero-Lara, Keita Teranishi, Sudip Seal

Organizations: Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

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.

Figures & tables

Panel of Figure 1 — Figure 1: The two rounding regimes. Both coordinate the same lower-or-upper scalar choices, but dynamic rounding re-rounds an observed activation row for each weight block it multiplies, whereas static rounding commits to one rounded weight matrix for a distribution of future inputs.

Panel of Figure 2 — Figure 2: The three comparison quantities. The constructive guarantee of Theorem 3.3 is additive relative to the binary optimum. Same-grid stochastic rounding has expectation at least the binary optimum; the dither expectation is unordered with both ( Section 2 ).

Symbol	Meaning
$K,p,n$	Shared inner dimension, dynamic block width, and number of columns in the reusable static factor.
$\ell,u,d_{k},\Delta$	Adjacent lower and upper levels, their coordinate-dependent step size, and a common step when one is assumed.
$\theta,\chi$	Observed offset fractions and binary lower/upper choices.
$U,F,D$	Active and fixed coordinate sets, and the embedding of active step sizes into the full inner dimension.
$e^{\mathrm{fix}},g^{\mathrm{fix}}$	Errors forced by saturation or exact representation, and their image through the weight block.
$W_{\mathrm{blk}},V,v_{k}$	Weight block, its step-weighted version with rows $v_{k}^{\top}$ , where $v_{k}=d_{k}(W_{\mathrm{blk}})_{k,:}^{\top}$ .

Table 1: Notation for the fixed-grid setup and the dynamic results; symbols for the static and hardness sections are defined where they first appear.

Figure 3: Null-space reduction preserves the relaxed product. This three-coordinate example uses

V=(1,1,1)^{\top}

, so the affine slice is

x_{1}+x_{2}+x_{3}=1.5

. Starting at

x^{0}=(0.3,0.5,0.7)^{\top}

, the first move reaches

x^{1}=(0.8,0,0.7)^{\top}

and fixes

x_{2}=0

on the highlighted face. The second move keeps

x_{2}=0

and fixes

x_{1}=1

, reaching

x^{\star}=(1,0,0.5)^{\top}

on the highlighted edge. Both moves preserve

V^{\top}x

, and the endpoint has one fractional coordinate, matching

\rank(V)=1

$r$	Exact	RTN	Bern.	Nearest	CE	Cert.	Diag. oracle	Face gap
$1$	$0.046$	$0.515$	$2.010$	$0.046$	$0.046$	$0.188$	$0.046$	$0$
$2$	$4.796\times 10^{-5}$	$0.655$	$2.013$	$0.088$	$0.088$	$0.375$	$0.088$	$0.084$
$4$	$0.009$	$0.940$	$2.007$	$0.188$	$0.165$	$0.750$	$0.144$	$0.136$
$8$	$0.130$	$0.837$	$1.971$	$0.407$	$0.347$	$1.500$	$0.255$	$0.120$

Table 2: Small-instance comparison at

K=16

. Method entries are medians across 10 block-level medians, with 25 targets per block. “Face gap” is the pooled median of (diagnostic-oracle error minus global exact error), normalized by dither. The positive gaps for

r=2,4,8

measure the difference between the best reached-face completion and the global optimum. Headings abbreviate round-to-nearest (RTN), same-grid Bernoulli (Bern.), conditional expectation (CE), and reached-face diagnostic oracle (Diag.). “Cert.” is

3r/K

. Each rank uses independent block draws, so the Exact column compares methods within a row; its cross-row variation is not a rank trend.

$K$	$r$	RTN	Bern.	Nearest	CE [IQR]	Diag. oracle
$64$	$4$	$0.794$	$2.025$	$0.045$	$0.042\,[0.038,0.045]$	$0.041$
$64$	$16$	$0.929$	$2.000$	$0.244$	$0.166\,[0.162,0.171]$	$0.114$
$256$	$4$	$0.827$	$1.991$	$0.012$	$0.011\,[0.009,0.012]$	$0.010$
$256$	$16$	$0.969$	$2.002$	$0.055$	$0.040\,[0.038,0.041]$	$0.027$
$1024$	$4$	$0.900$	$2.003$	$0.003$	$0.003\,[0.002,0.003]$	$0.003$
$1024$	$16$	$0.899$	$2.004$	$0.014$	$0.010\,[0.010,0.011]$	$0.007$

Table 3: Balanced scalable comparison for a representative subset of the swept

(K,r)

grid; the artifact reports all 25 configurations. Entries are medians across ten block-level medians; conditional-expectation (CE) completion also shows its interquartile range (IQR). Each block contributes 25 independent fractional targets. The Bernoulli column is an analytic expectation at each observed fractional target, whereas dither is the denominator and therefore equals one. RTN denotes round-to-nearest and Diag. the reached-face diagnostic oracle, which is exponential in

r

Method	Error/dither	Median time (ms)
RTN	$0.899$	$0.002$
Analytic Bernoulli evaluation	$2.004$	$0.015$
Nearest completion	$0.014$	$75.851$
CE completion	$0.010$	$75.916$
Reached-face diagnostic oracle	$0.007$	$76.961$

Table 4: Representative end-to-end time per fractional target at

K=1024,r=16

. Losses are medians across block medians; times are pooled medians and include null-space reduction and residual rounding. The Bernoulli timing measures closed-form evaluation of its expected loss. The reached-face diagnostic-oracle time is specific to

r=16

and retains exponential worst-case scaling. RTN denotes round-to-nearest and CE conditional expectation.

$K$	$p$	$r$	RTN/dither	CE/dither	Cert./dither	CE (raw)
$256$	$32$	$1$	$0.447$	$0.003$	$0.012$	$0.061$
$256$	$32$	$4$	$0.926$	$0.011$	$0.047$	$0.236$
$256$	$32$	$16$	$0.964$	$0.040$	$0.188$	$0.862$

Table 5: Fixed-width balanced blocks: ten independent blocks and 25 targets per block. Entries are medians across block medians. “Cert.” is the deterministic bound, and raw CE error is the squared Euclidean product error before normalization.

Figure 4: Balanced dynamic rounding. The left panel shows conditional-expectation (CE) completion against

K/r

by rank; colored bands are pooled target-level interquartile ranges. The right panel separates round-to-nearest (RTN), analytic Bernoulli, the two polynomial completion rules, and the reached-face diagnostic oracle; each right-panel point pools all

p=r

configurations with the displayed ratio. Its bands describe pooled heterogeneity across configurations, blocks, and targets; the tables report summaries for each configuration. Tables 2 – 3 report cross-block summaries.

$\alpha$	$\beta_{\rm rms}/\beta_{\max}$	Cert.	RTN	Bern.	Nearest	CE	Diag. oracle
$0$	$1.000$	$0.047$	$0.903$	$2.000$	$0.011$	$0.010$	$0.010$
$0.500$	$0.155$	$1.959$	$0.759$	$2.042$	$0.041$	$0.036$	$0.035$
$1.000$	$0.080$	$7.312$	$0.768$	$2.244$	$0.479$	$0.379$	$0.306$
$2.000$	$0.065$	$11.087$	$0.758$	$2.147$	$0.710$	$0.604$	$0.493$

Table 6: Row-norm sweep at

K=256,r=4

. “Cert.” is the deterministic conditional-expectation (CE) certified bound divided by dither; method columns are medians across the ten block-level medians. RTN denotes round-to-nearest, Bern. same-grid Bernoulli, Diag. the reached-face diagnostic oracle, and RMS root mean square. All blocks have unit RMS row norm.

$\rho$ (%)	$\theta$ /dither	QP/dither	$\theta$ (raw)	QP (raw)	Ratio	QP time (ms)
$0$	$0.010$	$0.009$	$0.215$	$0.199$	$1.1$	$13.421$
$5$	$0.181$	$0.010$	$3.853$	$0.213$	$18.1$	$12.686$
$10$	$0.450$	$0.010$	$9.597$	$0.208$	$46.1$	$12.131$

Table 7: Fixed-offset study. Error columns are medians across ten block-level medians, each based on 25 targets; improvement is their ratio. The dither denominator is

256/12

throughout. Runtime includes the continuous-relaxation solve. QP denotes the box-constrained quadratic program.

Panel of Figure 5 — Figure 5: Stress tests for the dynamic-rounding construction. (a) Measured conditional-expectation (CE) error and its certified bound under increasing row-norm imbalance. (b) Clipping-aware quadratic-program (QP) initialization keeps the product error small as the fixed-coordinate fraction grows. Each estimate pools ten blocks and 25 targets per block; vertical bars in (b) span pooled interquartile ranges.

Bits	Cal.	Method	Held-out/RTN [95% RI]
$3$	128	RTN	$1.000\,[1.000,1.000]$
$3$	128	RTN + bias	$0.860\,[0.830,0.871]$
$3$	128	Centered, fixed bias	$1.264\,[0.933,2.035]$
$3$	128	Centered + bias	$0.756\,[0.598,0.837]$
$3$	128	Uncentered	$0.842\,[0.692,0.929]$
$3$	full	RTN	$1.000\,[1.000,1.000]$

Table 8: Held-out product mean-squared-error (MSE) ratios on Digits, normalized by split-matched round-to-nearest (RTN) at the same precision. Entries are medians of 5 split-level medians; the bracketed intervals are bootstrap 95% resampling intervals (RI) over those five observed splits. For calibration size 128, each split-level median summarizes 20 subsets; “full” uses the complete calibration pool once per split. RTN+bias and centered+bias use their own errors with the same calibration mean.

Figure 6: Generalization from calibration to held-out data across five Digits splits. Both axes use the split-matched round-to-nearest (RTN) denominator. Marker shapes distinguish RTN-plus-bias, centered fixed-bias, centered-plus-bias, and uncentered rounding; filled markers show 128-example subsets and open markers show the full calibration pool.

Method / role	Target	Loss or norm	Rounding choices	Reuse constraint
Dynamic construction	One observed row–block product	Squared Euclidean	Fixed adjacent levels; forced clipping retained	One activation copy per output block
Static identity	Expected linear-layer output	Second-moment quadratic	Fixed columnwise sets	One weight matrix for future inputs
AdaRound Nagel et al.,2020	Calibration layer outputs	Reconstruction loss	Learned adjacent-level decisions	Weights reused across inputs
GPTQ Frantar et al.,2023	Calibration linear outputs	Gram-weighted quadratic	Grid rounding with sequential compensation	Weights reused across inputs
Linear discrepancy Lovász et al.,1986	Worst fractional target	Commonly $\ell_{\infty}$	Binary vector	Guarantee over targets
Babai Babai,1986	Nearest lattice point	Euclidean distance	Unrestricted integer basis coefficients	Fixed lattice basis; new target per query

Table 9: Comparison of mathematical targets and feasible sets. The static identity specifies a loss; AdaRound and GPTQ supply optimization procedures. Babai’s lattice coefficients require additional constraints to represent the adjacent-level choices of this paper.

Figure 7: Tail operators divided by the dither variance. Colored markers are empirical Gram–Schmidt medians for flat (

\lambda_{j}=1

), polynomial (

\lambda_{j}=1/j

), and geometric (

\lambda_{j}=0.75^{j-1}

) spectra. The gray diamonds give the analytic independent same-grid Bernoulli control for the same eigenbases and fractional targets. Bars span interquartile ranges across four paired instances, and each Gram–Schmidt estimate uses 1,024 outputs. The horizontal dashed line is the dither scale

\Delta^{2}/12

Bits	Cal.	Paired comparison	Ratio [95% RI]	Mean	Wins
$3$	128	Centered / RTN	$1.264\,[0.933,2.035]$	$1.325$	$1/5$
$3$	128	Centered+bias / RTN+bias	$0.869\,[0.721,0.973]$	$0.863$	$5/5$
$3$	full	Centered / RTN	$1.250\,[0.884,2.027]$	$1.298$	$1/5$
$3$	full	Centered+bias / RTN+bias	$0.842\,[0.694,0.946]$	$0.827$	$5/5$
$4$	128	Centered / RTN	$1.148\,[0.867,1.753]$	$1.314$	$1/5$
$4$	128	Centered+bias / RTN+bias	$0.667\,[0.615,0.687]$	$0.656$	$5/5$

Table 10: Paired held-out product MSE comparisons on Digits. Ratios compare the methods shown in each row, so corrected weights are compared with corrected RTN. RI denotes a bootstrap resampling interval over five split-level median ratios.

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Contraction-Gauge Preconditioning for Quantized Matrix Multiplication

Piyush Sao, Narasinga Miniskar, Pedro Valero-Lara, Keita Teranishi+1

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

Post-Training Quantization Low-Bit Quantization Neural Network Quantization

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We study low-precision computation of C=AB with both factors quantized. We derive an exact finite-dimensional identity for the expected squared product error under independent, zero-mean entrywise errors with known variance fields; it holds exactly for non-overloading subtractive dither and for independent stochastic rounding, and we empirically assess deterministic round-to-nearest (RTN). Using the product-preserving equivalence AB=(AT)(T^{-1}B), we formulate contraction-gauge preconditioning: jointly choosing a factor representation and its sharing pattern before quantization. Preconditioning can reduce product error but may require extra transformed, quantized copies of the opposite operand: a shared transform needs one copy, a block-specific transform up to one per block. Within the bounded family of positive diagonal gauges (folds), a geometric program computes a globally optimal shared fold and a linear program decides whether the identity fold is already optimal. For other families we derive computable selection statistics -- tail index for scaling, profile spread for partitioning, coherence and weighted-Gram energy for rotations, slice-energy covariance for hierarchy depth -- with upper bounds for ranking heuristic candidates. Across twelve linear products from a trained three-block image classifier, median within-product rank correlations between dither-model predictions and deterministic-RTN errors are 0.937 at 8 bits and 0.918 at 4 bits. The GP fold cuts held-out product error over the identity fold by 18.0% (8-bit) and 20.5% (4-bit) in geometric mean, beats a SmoothQuant-style grid baseline at both precisions and on ten of twelve products, and lowers composed logit MSE by 15.4% and 26.4%. We thus provide exact stochastic product-error accounting, certified selection within the diagonal family, and a common objective for evaluating reusable transform candidates under RTN.

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