Measurement-Based Rounding Scheme

• Measurement-based rounding scheme is a framework where auxiliary observations drive discrete decisions beyond simple nearest-grid rounding.
• It spans geometric partitions, distributed algorithms, post-training quantization, and LWR cryptography, adapting measurements like local neighborhoods, potentials, and reconstruction loss.
• The approach guarantees performance bounds and consistency by preserving local invariants under resource constraints such as limited side information and bounded communication.

Measurement-based rounding scheme denotes a class of procedures in which a discrete rounding decision is driven by an auxiliary measurement rather than by nearest-grid distance alone. In the cited literature, that measurement takes several technically distinct forms: a local $\ell_\infty$ neighborhood in a partition of $\mathbb{R}^d$, a pairwise-decomposed potential on a graph, a calibration-set reconstruction loss for post-training quantization, a small side-information message used to synchronize repeated Euclidean measurements, or a fixed threshold test implemented by add-and-shift quantizers in Learning-with-Rounding cryptography (Woude et al., 2022, Faour et al., 2022, Nagel et al., 2020, Lee et al., 2023, Dunkelman et al., 2020, Kundu et al., 2024). The common structural feature is that rounding is constrained by what can be inferred from a local observation, a bounded message, or a task-specific measurement of downstream effect.

1. Conceptual scope

The phrase encompasses multiple formal models rather than a single canonical algorithm. In geometric rounding, a deterministic rounding map is equivalent to a partition of $\mathbb{R}^d$; the relevant measurement is the local neighborhood $B_\epsilon(p)$ and the key question is how many partition cells that neighborhood can intersect. In distributed graph algorithms, the measurement is a potential of the form $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$, and rounding proceeds by local updates that preserve this potential approximately. In post-training quantization, the measurement is the discrepancy between the original and quantized layer or block outputs on a calibration set. In consistent rounding with side information, the measurement is a first noisy observation together with a short message identifying a color class. In LWR-based cryptography, the measurement is a residue compared against fixed thresholds, realized by additions of half-step constants and power-of-two shifts (Woude et al., 2022, Faour et al., 2022, Nagel et al., 2020, Lee et al., 2023, Dunkelman et al., 2020, Kundu et al., 2024).

Setting	Measurement guiding rounding	Rounding mechanism
Euclidean partitions	$B_\epsilon(p)$ and $N_\epsilon(p)$	choose a representative of the intersected cell
Distributed local rounding	$\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$	local label updates under defective colorings
Post-training quantization	layer/block reconstruction loss on calibration data	learn rounding of weights to a quantization grid
Consistent rounding with side information	color $S(X_1)$ of the first measurement’s tile	round the second measurement to the nearest center in that color class
LWR-based reconciliation	threshold tests via $h_1,h_2,h_3$ and shifts	add-half and right-shift quantization

This taxonomy suggests that “measurement-based” should be read operationally: rounding is not purely a function of the current scalar to be discretized, but of a structured observation that encodes local geometry, pairwise interactions, output reconstruction, reconciliation state, or hardware-friendly threshold information.

2. Partition-based measurement in Euclidean space

A central geometric formalization identifies deterministic rounding and partitions as equivalent. A deterministic rounding scheme in $\mathbb{R}^d$0 is a function $\mathbb{R}^d$1, which induces the partition

$\mathbb{R}^d$2

Conversely, a partition $\mathbb{R}^d$3 with representatives $\mathbb{R}^d$4 induces a rounding map $\mathbb{R}^d$5 for $\mathbb{R}^d$6. The measurement-based requirement is then expressed through a seclusion property: for the closed $\mathbb{R}^d$7 ball

$\mathbb{R}^d$8

one defines

$\mathbb{R}^d$9

and asks that $\mathbb{R}^d$0 for all $\mathbb{R}^d$1. For partitions into unit hypercubes, this is the secluded hypercube partition problem (Woude et al., 2022).

The paper gives an explicit solution via “reclusive partitions.” For a reclusive matrix $\mathbb{R}^d$2, one defines the lattice $\mathbb{R}^d$3 and the partition

$\mathbb{R}^d$4

Adjacency is characterized by weak-alt-1 sequences in $\mathbb{R}^d$5, and nonadjacent cubes are separated by a positive reclusive distance $\mathbb{R}^d$6. An explicit coloring

$\mathbb{R}^d$7

shows that the partition graph is $\mathbb{R}^d$8-colorable, hence every clique has size at most $\mathbb{R}^d$9. Choosing $B_\epsilon(p)$0 so that $B_\epsilon(p)$1 yields the hypercube partition theorem:

$B_\epsilon(p)$2

The construction is explicit and efficiently computable.

The same work establishes sharp lower bounds for broad deterministic classes. For any unit hypercube partition of $B_\epsilon(p)$3, there exists a point $B_\epsilon(p)$4 with $B_\epsilon(p)$5, so no deterministic unit-hypercube scheme can achieve degree $B_\epsilon(p)$6. Stronger optimality theorems extend this obstruction to partitions with bounded measure or bounded diameter. A universal tolerance bound further shows that if each cell has strict pairwise bound $B_\epsilon(p)$7 and $B_\epsilon(p)$8 for all $B_\epsilon(p)$9, then

$\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$0

and, using $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$1,

$\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$2

The constructive value $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$3 is therefore within a factor $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$4 of the universal obstruction. A common misconception is that deterministic local rounding might force fewer than $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$5 outcomes everywhere by a sufficiently clever tiling; the degree optimality results rule this out for the paper’s “reasonable” classes of partitions.

3. Consistent rounding with side information

A distinct measurement-based model studies repeated noisy measurements $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$6 under the $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$7 norm. Space is partitioned into connected, closed, bounded tiles of volume at most $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$8, each tile is assigned a center and a color in $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$9, and the first measurement sends only the color of its tile. If $B_\epsilon(p)$0 denotes the tiles of color $B_\epsilon(p)$1 and $B_\epsilon(p)$2 their centers, the second measurement rounds according to

$B_\epsilon(p)$3

The central notion is fault tolerance. If $B_\epsilon(p)$4 is the minimal distance between same-colored points in different tiles, then the fault tolerance radius is $B_\epsilon(p)$5, equivalently the largest $B_\epsilon(p)$6 such that the $B_\epsilon(p)$7-inflations of same-colored tiles are pairwise disjoint. The consistency requirement is

$B_\epsilon(p)$8

for all $B_\epsilon(p)$9 (Dunkelman et al., 2020).

The minimal side-information threshold is exact: positive fault tolerance in $N_\epsilon(p)$0 requires at least $N_\epsilon(p)$1 colors, and there is a construction with $N_\epsilon(p)$2 colors achieving $N_\epsilon(p)$3. Hence $N_\epsilon(p)$4 bits are necessary and sufficient in the worst case. Necessity is proved geometrically for polytopic tiles and topologically for contractible tiles via Čech and de Rham cohomology; sufficiency comes from an explicit dimension-reducing construction.

With more colors, the achievable tolerance scales like $N_\epsilon(p)$5 up to constants. The paper derives upper bounds from Brunn–Minkowski and sphere packing. In dimension three, optimal sphere packing yields the tight asymptotic law

$N_\epsilon(p)$6

In dimension two,

$N_\epsilon(p)$7

and this is matched by an explicit honeycomb-of-rectangles construction. For $N_\epsilon(p)$8 and $N_\epsilon(p)$9, the constants are $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$0 and $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$1, respectively. This model makes explicit a tradeoff absent from ordinary nearest rounding: small unidirectional communication can convert a discontinuous quantizer into a perfectly consistent one on a prescribed neighborhood.

4. Local distributed rounding via measured potentials

In deterministic distributed algorithms, measurement-based rounding appears as local derandomization of fractional or probabilistic label assignments. The setting is an undirected graph $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$2 with maximum degree $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$3, under the LOCAL or CONGEST model. Each node chooses a label from a finite alphabet $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$4, and the analysis is organized around a pairwise-decomposed potential

$\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$5

The framework separates this into nonnegative utility $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$6 and nonnegative cost $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$7, both decomposable over an edge set $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$8 of an auxiliary multigraph $\Phi_i(\lambda)=u(\lambda)-\eta_i c(\lambda)$9, and performs deterministic rounding while maintaining

$S(X_1)$0

with a scheduled coefficient $S(X_1)$1 (Faour et al., 2022).

The basic rounding step starts from a $S(X_1)$2-integral fractional assignment, computes edge weights $S(X_1)$3, finds a weighted $S(X_1)$4-relative average defective $S(X_1)$5-coloring with $S(X_1)$6, and then processes one color class at a time. A node locally compares candidate labels using edge-summed quantities

$S(X_1)$7

updates its fractional probabilities by $S(X_1)$8, and doubles the integrality granularity. The key guarantee is

$S(X_1)$9

and after $h_1,h_2,h_3$0 such steps,

$h_1,h_2,h_3$1

This abstraction covers maximal independent set, maximum-weight independent set approximation, minimum-cost set cover approximation, matching, hypergraph matching, and coloring. As a highlighted consequence, the framework yields a deterministic $h_1,h_2,h_3$2-round MIS algorithm in the LOCAL model and an $h_1,h_2,h_3$3-round deterministic MIS algorithm in the CONGEST model. It also places MIS, maximal matching, $h_1,h_2,h_3$4-vertex coloring, and $h_1,h_2,h_3$5-edge coloring in the same $h_1,h_2,h_3$6 deterministic regime. The measurement here is not spatial but analytic: a local rounding move is accepted because it preserves a globally meaningful potential whose decomposition is pairwise and locality-compatible.

5. Measurement-based rounding in post-training quantization

In post-training quantization, measurement-based rounding learns discrete quantization decisions by minimizing output discrepancy on a small calibration set rather than minimizing parameter-space error. For a layer or block $h_1,h_2,h_3$7 with pre-trained weights $h_1,h_2,h_3$8 and quantized weights $h_1,h_2,h_3$9, the canonical objective is

$\mathbb{R}^d$00

or, in the formulations used for AdaRound and FlexRound, a Frobenius-norm reconstruction loss on layer or block outputs (Nagel et al., 2020, Lee et al., 2023).

AdaRound formalizes per-weight up/down decisions as binary variables. With scale $\mathbb{R}^d$01 and clipping range $\mathbb{R}^d$02, round-to-nearest is

$\mathbb{R}^d$03

whereas AdaRound chooses, for each weight, whether to use floor or ceil:

$\mathbb{R}^d$04

A second-order Taylor approximation of the task loss leads to a QUBO in the rounding variables. The intractable Hessian objective is then replaced by a tractable layer-wise surrogate,

$\mathbb{R}^d$05

and, more accurately, by an asymmetric objective using measured float inputs $\mathbb{R}^d$06 and quantized-prefix inputs $\mathbb{R}^d$07:

$\mathbb{R}^d$08

The soft rounding variable is

$\mathbb{R}^d$09

with $\mathbb{R}^d$10 and $\mathbb{R}^d$11, and the regularizer is

$\mathbb{R}^d$12

Using 4-bit weights and FP32 activations, AdaRound reports 68.71$\mathbb{R}^d$130.06 for ResNet-18 and 75.23$\mathbb{R}^d$140.04 for ResNet-50, while round-to-nearest gives 23.99 and 35.60 under the same weight precision; for DeeplabV3+ on Pascal VOC, 4-bit W and 8-bit A reaches 70.86$\mathbb{R}^d$150.37 mIOU versus 6.09 for round-to-nearest at 4/8.

FlexRound retains the measurement-by-reconstruction principle but replaces additive learnable offsets with a division-based parameterization. Its quantization operator is

$\mathbb{R}^d$16

with $\mathbb{R}^d$17 factorized into a common grid size $\mathbb{R}^d$18 and positive learnable components such as $\mathbb{R}^d$19, depending on layer type. For a linear layer,

$\mathbb{R}^d$20

Under the straight-through estimator, the “reciprocal rule” yields

$\mathbb{R}^d$21

so scale updates are explicitly modulated by pre-trained weight magnitude. The measured objective is again reconstruction,

$\mathbb{R}^d$22

Empirically, FlexRound improves 4-bit weight-only MobileNetV2 top-1 accuracy from 69.46 to 70.82 relative to AdaRound, improves 2-bit ResNet-50 from 48.47 to 63.67, and quantizes LLaMA-33B to 8-bit with WikiText2 perplexity 6.82 versus 6.35 for the half-precision baseline. A notable ablation shows that fixing $\mathbb{R}^d$23 makes FlexRound perform similarly to AdaRound, indicating that joint learning of the common grid size is integral to the method.

The PTQ literature therefore uses “measurement-based” in a task-aware sense: the quantizer is optimized against observed output reconstruction on calibration data. This sharply differs from deterministic nearest rounding, which is blind to inter-weight interactions and downstream loss.

6. Threshold-based rounding and reconciliation in LWR cryptography

In Learning-with-Rounding cryptography, rounding is a deterministic substitute for explicit error sampling. Scabbard studies three KEMs—Florete, Espada, and Sable—built from RLWR or MLWR, all using power-of-two moduli and hardware-friendly quantizers. If $\mathbb{R}^d$24 and $\mathbb{R}^d$25, the coefficient-wise map from modulus $\mathbb{R}^d$26 to modulus $\mathbb{R}^d$27 is

$\mathbb{R}^d$28

which is exactly $\mathbb{R}^d$29 implemented by add-half and right-shift. Encryption and decryption use helper constants $\mathbb{R}^d$30 and fixed shifts to perform rounding, message embedding, and recovery without division (Kundu et al., 2024).

The generic core is:

• key generation computes $\mathbb{R}^d$31;
• encryption computes $\mathbb{R}^d$32 and a compressed helper $\mathbb{R}^d$33 after subtracting the embedded message amplitude;
• decryption recomputes a nearby value $\mathbb{R}^d$34 and recovers the message by

$\mathbb{R}^d$35

Correctness is governed by the decryption noise $\mathbb{R}^d$36. The stated sufficient condition for no decryption failure is

$\mathbb{R}^d$37

The three schemes differ primarily in how they organize the same threshold structure. Florete is RLWR-based with $\mathbb{R}^d$38 and message repetition across the polynomial, followed by majority decoding in original_msg; the paper reports $\mathbb{R}^d$39 for the three security levels. Espada is MLWR-based with very small $\mathbb{R}^d$40 and packs $\mathbb{R}^d$41 bits per coefficient, with reported $\mathbb{R}^d$42. Sable is Saber-like MLWR with reduced moduli and smaller secrets, reporting $\mathbb{R}^d$43. In all cases, the measurement is a threshold decision on a noisy modular value, and the helper data $\mathbb{R}^d$44 identifies the quantization bin so that the receiver can invert the embedding.

Scabbard’s hardware study treats rounding and reconciliation as effectively negligible relative to polynomial multiplication and Keccak, precisely because the quantizers are add-and-shift. On Cortex-M4, Florete low/medium/high key generation takes 299k/439k/598k cycles, encapsulation 536k/815k/1131k, and decapsulation 606k/957k/1357k; Sable low/medium/high uses 381k/745k/1251k, 558k/1005k/1593k, and 568k/1031k/1622k cycles for key generation, encapsulation, and decapsulation, respectively. On UltraScale+, Sable reaches 31.6/39.35/48.43 $\mathbb{R}^d$45s for key generation, encapsulation, and decapsulation at 150 MHz. The design rationale is explicitly hardware-aware: power-of-two quantizers avoid divisions, simplify masking, and eliminate timing issues associated with prime-modulus division.

7. Optimality, limitations, and open directions

Across these literatures, measurement-based rounding is constrained by sharp lower bounds and domain-specific scope conditions. In geometric partition rounding, $\mathbb{R}^d$46 is optimal for unit hypercube partitions and, more generally, for bounded-measure and bounded-diameter classes; the main open question is whether the optimal tolerance with $\mathbb{R}^d$47 is $\mathbb{R}^d$48, since the explicit construction gives $\mathbb{R}^d$49 while the current universal bound is $\mathbb{R}^d$50. In consistent rounding with side information, exact asymptotic constants are known in dimensions $\mathbb{R}^d$51, but improving constants in other dimensions and optimizing small-$\mathbb{R}^d$52 constructions remain open. In distributed local rounding, the framework applies to problems whose randomized analyses reduce to pairwise utilities and costs; higher-order dependencies fall outside the present abstraction, and randomized algorithms may still have better asymptotic running times on some tasks. In PTQ, very low bit-widths remain difficult without fine-tuning, calibration mismatch can degrade performance, and the continuous relaxations used by AdaRound and FlexRound do not solve the underlying discrete optimization globally. In LWR cryptography, shrinking $\mathbb{R}^d$53 and $\mathbb{R}^d$54 improves bandwidth and hardware cost but tightens quantization bins, so correctness must be protected by carefully balancing secret distributions, repetition or packing strategies, and the parameters $\mathbb{R}^d$55 and $\mathbb{R}^d$56 (Woude et al., 2022, Dunkelman et al., 2020, Faour et al., 2022, Nagel et al., 2020, Lee et al., 2023, Kundu et al., 2024).

A plausible implication is that the phrase “measurement-based rounding scheme” is best understood as a unifying methodological description rather than a single theory. What unifies the cited work is a design principle: discrete outputs are selected to preserve a measured local invariant—geometric seclusion, consistency radius, pairwise potential, output reconstruction fidelity, or reconciliation correctness—under explicit resource constraints such as bounded neighborhood degree, limited side information, local communication, calibration size, or hardware cost.