Pairwise Interaction Sampling

• Pairwise interaction sampling is a method where the basic unit is a pair, addressing the quadratic explosion in comparisons for tasks like preference aggregation and testing.
• It underpins active selection strategies such as Hybrid-MST and ASAP, enhancing latent score recovery and performance metrics in ranking and recommendation systems.
• The concept extends to exact and approximate sampling in stochastic systems and combinatorial test generation, using specialized algorithms to optimize pair coverage and estimation variance.

Searching arXiv for recent and foundational work on pairwise interaction sampling across preference aggregation, pairwise learning, exact sampling, and software testing. Pairwise interaction sampling denotes a family of methods in which the primary unit of acquisition, computation, or coverage is a pair rather than an individual object. Across current literature, the term appears in at least four closely related senses: active selection of pairwise comparisons for latent-score recovery in preference aggregation and subjective assessment, direct subsampling of pairwise losses for learning and optimization, exact or approximate sampling in stochastic systems whose law is defined by pairwise interactions, and combinatorial generation of configuration samples that cover all feasible feature pairs in software testing. The common structural driver is the quadratic growth of the pair universe—typically $\frac{n(n-1)}{2}$ or $\binom{n}{2}$—which makes exhaustive enumeration, comparison, or evaluation impractical and shifts attention to principled pair selection, weighting, or coverage design.

1. Conceptual scope and formal problem classes

In preference aggregation and subjective assessment, the basic object is a set of items $A_1,\dots,A_n$ or stimuli, together with noisy binary outcomes indicating which element in a queried pair is preferred. The observable data are stored in a pairwise comparison matrix such as $\mathbf{M}=(m_{ij})$ or $PCM[i,j]$, and the inferential target is a latent score or ranking recovered from incomplete pairwise evidence. The central computational obstacle is that a complete pairwise design requires all possible pairs, so the number of comparisons grows quadratically with the number of stimuli (Li et al., 2018, Mohammadi et al., 2023).

In pairwise learning, the basic object is not a comparison matrix but an empirical order-2 U-statistic. The canonical form is

$$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$

which appears in ranking, metric learning, clustering, contrastive representation learning, graph embedding, and face verification. Here the difficulty is again quadratic cost, but the statistically relevant fact is that the sampling unit is the pair population $\mathcal J=\{(i,j):1\le i<j\le N\}$, not merely the observation set (Davy et al., 1 Jun 2026).

In software-testing formulations, the pairwise unit is a feature interaction rather than a statistical observation. A propositional feature model $\varphi(x_1,\dots,x_n)$ defines valid configurations, and the goal is to construct a minimum-cardinality set of configurations such that every feasible pairwise interaction of concrete literals is covered at least once. In this setting, “sampling” refers to test-suite construction under feasibility constraints rather than stochastic subsampling (Fekete et al., 7 Oct 2025).

In stochastic geometry and exact-sampling theory, pairwise interaction sampling refers to simulation from laws whose density or potential is determined by pairwise terms. The pairwise interaction structure may define a Gibbs point process, a Markov chain induced by pairwise comparisons, or a finite-volume Gibbs measure with pair interaction. The operational question is then not which pairs to query, but how to generate exact or controlled-approximation samples from the resulting distribution (Moka et al., 2019, Fotakis et al., 2022, Pra et al., 2012).

2. Active pair selection in preference aggregation and subjective assessment

The most developed statistical use of pairwise interaction sampling is active pair selection for latent-score recovery. In the Bradley–Terry formulation used by Hybrid-MST, if $s_i$ and $s_j$ are latent scores, then the comparison probability is

$\binom{n}{2}$0

After maximum-likelihood estimation, the score vector is approximated locally by a Gaussian posterior, and the score difference $\binom{n}{2}$1 is given a pair-specific Gaussian approximation. Hybrid-MST defines the utility of a candidate pair as its expected information gain, estimated efficiently by Gaussian–Hermite quadrature with $\binom{n}{2}$2 sample points. The method then switches between a sequential Global Maximum rule and a batch Minimum Spanning Tree rule with edge weights $\binom{n}{2}$3, using Global Maximum while $\binom{n}{2}$4 and Minimum Spanning Tree sampling thereafter (Li et al., 2018).

That hybridization addresses a structural tension in active comparison design. Sequential EIG maximization is information-efficient in sparse regimes but poorly matched to crowd annotation or parallel experimentation. The MST phase produces exactly $\binom{n}{2}$5 comparisons on $\binom{n}{2}$6 vertices, enforces global connectivity, and yields a batch of high-information edges. This makes the method simultaneously an active-learning policy and a graph-design heuristic for stable Bradley–Terry recovery. The empirical pattern reported for simulated, VQA, and IQA data is that Hybrid-MST is especially strong for rating aggregation, while some alternatives can be slightly better for pure ranking only at very small budgets (Li et al., 2018).

ASAP generalizes the same EIG logic but changes the inference layer. Instead of partial or local posterior updates, it performs full posterior updates via approximate message passing on a factor graph, with Gaussian approximation obtained through expectation propagation or moment matching. To keep the repeated EIG calculations feasible, ASAP evaluates information gain only on a subset of candidate pairs selected by the uncertainty proxy

$\binom{n}{2}$7

normalized per condition, and supports batch selection through an MST over inverse information gain. The reported effect is improved RMSE and competitive or superior SROCC relative to Hybrid-MST, Crowd-BT, HR-active, and sorting-based methods, including on large-scale simulations; for 200 conditions, ASAP reaches RMSE $\binom{n}{2}$8 in about $\binom{n}{2}$9 standard trials (Mikhailiuk et al., 2020).

A separate evaluation framework for subjective image and video assessment makes the same point from a benchmarking perspective. It initializes $A_1,\dots,A_n$0 with all $A_1,\dots,A_n$1s, simulates incomplete experiments by repeatedly selecting pairs and drawing judgments from a complete dataset, and compares incomplete-design Bradley–Terry scores with full-design scores via PLCC and SROCC. Within that framework, active methods such as HR-active, ASAP, and Hybrid-MST outperform random and simple sorting-based strategies; with only $A_1,\dots,A_n$2 of all possible pairs, active methods already achieve correlations greater than $A_1,\dots,A_n$3, and at around $A_1,\dots,A_n$4 of the pairs most methods reach correlations close to $A_1,\dots,A_n$5 (Mohammadi et al., 2023).

A recurrent misconception is that pairwise tests require near-complete pair coverage to remain reliable. The comparative evidence does not support that claim. The published results instead indicate that the informativeness of the queried subset matters more than raw pair count, especially when the objective is stable recovery of psychometric scores rather than exhaustive observation of every pair (Mohammadi et al., 2023).

3. Direct pair sampling for learning, estimation, and randomized optimization

In empirical pairwise loss estimation, the decisive methodological question is whether sampling should be performed on observations or directly on pairs. The survey-sampling approach of direct pair selection treats the pair population $A_1,\dots,A_n$6 as the population of interest and uses a Horvitz–Thompson estimator

$A_1,\dots,A_n$7

where $A_1,\dots,A_n$8 is the inclusion probability of pair $A_1,\dots,A_n$9. Under Poisson pair sampling, the optimal direct design assigns pair inclusion probabilities proportional to

$\mathbf{M}=(m_{ij})$0

and the paper proves that, for the same expected number of evaluated pairs, direct pair sampling strictly dominates observation sampling in variance under a non-factorization condition (Davy et al., 1 Jun 2026).

This result is conceptually important because it separates “pairwise” from “induced by a sample of points.” Observation sampling constrains pair inclusion probabilities to factorize as $\mathbf{M}=(m_{ij})$1; direct pair sampling does not. In problems where pair informativeness depends on genuinely pair-level structure, that restriction is costly. The theoretical and experimental message is therefore sharp: if the objective is pairwise, the design should target pairs directly. This principle is reinforced by reported variance reductions on Toy, MovieLens, Cora, and LFW, and by the result that when the number of sampled pairs scales linearly, $\mathbf{M}=(m_{ij})$2, the survey-weighted empirical minimizer attains the same $\mathbf{M}=(m_{ij})$3 statistical rate as full pairwise learning (Davy et al., 1 Jun 2026).

A complementary line of work analyzes adaptive pair sampling within stochastic optimization. In pairwise SGD and pairwise SGDA, each update is driven by a sampled pair $\mathbf{M}=(m_{ij})$4, and the empirical objective is again a U-statistic rather than a sum of independent terms. The PAC-Bayes analysis of randomized pairwise learning models the entire index-sampling sequence as a random hyperparameter $\mathbf{M}=(m_{ij})$5, uses a sample-independent prior $\mathbf{M}=(m_{ij})$6 and a sample-dependent posterior $\mathbf{M}=(m_{ij})$7, and derives generalization bounds controlled by $\mathbf{M}=(m_{ij})$8. The key significance is that the analysis accommodates arbitrary data-adaptive sampling schemes, rather than only uniform sampling, while explicitly handling the dependence structure induced by pair reuse (Zhou et al., 3 Apr 2025).

This suggests a precise distinction between two design problems that are often conflated. One problem is statistical efficiency of the sampled pair set, addressed by direct pair-level inclusion probabilities and Horvitz–Thompson correction. The other is generalization control for algorithms whose sampling policy is itself adaptive, addressed by stability-plus-PAC-Bayes bounds over discrete pair-index distributions. The former optimizes estimator variance; the latter quantifies the price of adaptive sampling through $\mathbf{M}=(m_{ij})$9 (Davy et al., 1 Jun 2026, Zhou et al., 3 Apr 2025).

4. Utility-based elicitation and decision-focused pair queries

Not all active pairwise sampling is uncertainty sampling. In personalized recommendation, the active query need not be the pair with the most uncertain outcome; it may instead be the pair whose answer is expected to improve the final decision the most. The utility-based elicitation framework formalizes this by defining an application-specific utility $PCM[i,j]$0, expected utility

$PCM[i,j]$1

and a query rule that selects the pair expected to produce the largest improvement in the utility of the final recommended outcome after posterior updating (Boroomand et al., 12 Aug 2025).

The preference model is Plackett–Luce. For pairwise data,

$PCM[i,j]$2

with log-scores $PCM[i,j]$3. The learning objective is the pairwise log-likelihood, and the scoring model is interchangeable: the paper instantiates it once with matrix factorization $PCM[i,j]$4 and once with a NeuMF-style neural recommender. The active criterion is then utility-aligned rather than uncertainty-aligned, and exact evaluation is approximated by Monte Carlo sampling from the ranking distribution (Boroomand et al., 12 Aug 2025).

This design changes the interpretation of informativeness. In standard EIG-based sampling, a pair is informative when it is uncertain and likely to alter the posterior. In utility-based elicitation, a pair is informative only if its answer is expected to change the quality of the downstream menu or selection. The distinction is explicit in the paper’s experiments: on MovieLens 100k, utility-based active sampling improves top-$PCM[i,j]$5 recommendation quality over random sampling and no additional sampling, and in the admissions setting it outperforms entropy-based uncertainty sampling, random sampling, and cluster-based sampling on Precision@10 and NDCG@10 (Boroomand et al., 12 Aug 2025).

A plausible implication is that “best pair” is application-dependent even when the underlying pairwise preference model is fixed. Once the downstream decision is encoded in $PCM[i,j]$6, the pairwise sampling problem becomes a decision-theoretic design problem rather than a purely inferential one (Boroomand et al., 12 Aug 2025).

5. Exact and approximate sampling under pairwise interaction laws

In exact sampling from pairwise comparisons, the observable primitive is a local comparison oracle rather than direct access to the target distribution $PCM[i,j]$7. The Local Sampling Scheme returns $PCM[i,j]$8, with $PCM[i,j]$9 and $$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$0. In the pairwise case, $$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$1, and the induced Markov chain has transition matrix

$$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$2

whose stationary distribution is exactly $$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$3. Coupling from the Past therefore gives perfect samples from $$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$4, but the sample complexity can depend disastrously on the geometry of $$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$5; in some natural scenarios, the naive method can be exponentially slow (Fotakis et al., 2022).

The main algorithmic remedy is to learn an approximation $$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$6 from pairwise comparisons, reweight the chain so that it becomes nearly uniform, and then apply CFTP plus rejection correction. Under identifiability and an efficiently learnable condition, the resulting exact sampler uses $$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$7 samples, with a high-probability refinement of $$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$8 (Fotakis et al., 2022). This establishes that exact sampling from pairwise comparison data is feasible without structural dependence on the target distribution’s multimodality.

A distinct exact-sampling literature treats Gibbs point processes with finite-range pairwise interaction potential

$$\widehat{U}_N(\theta)=\frac{2}{N(N-1)}\sum_{1\le i<j\le N}\ell(\theta,(Z_i,Z_j)),$$9

where $\mathcal J=\{(i,j):1\le i<j\le N\}$0 whenever $\mathcal J=\{(i,j):1\le i<j\le N\}$1. Partial rejection sampling rewrites the target law as a product distribution conditioned on the absence of local bad events across neighboring cells, resampling only locally implicated variables. For interaction range $\mathcal J=\{(i,j):1\le i<j\le N\}$2, the expected number of PRS iterations is $\mathcal J=\{(i,j):1\le i<j\le N\}$3, and with distributive or parallel implementation the expected running time is also $\mathcal J=\{(i,j):1\le i<j\le N\}$4, provided the intensity is not too high (Moka et al., 2019).

The saturated pairwise interaction Gibbs point process provides another exact-sampling construction, now in multitype ecology. Its Papangelou conditional intensity

$\mathcal J=\{(i,j):1\le i<j\le N\}$5

is locally stable when all medium-range coefficients satisfy $\mathcal J=\{(i,j):1\le i<j\le N\}$6, with explicit upper bound

$\mathcal J=\{(i,j):1\le i<j\le N\}$7

That local stability enables coupling from the past; when the condition is inconvenient, the paper uses an unconditional Metropolis–Hastings birth–death algorithm instead (Flint et al., 2022).

Approximate sampling from pair-interaction Gibbs measures can also be organized around parallel dynamics rather than exact coalescence. The Probabilistic Cellular Automaton construction introduces a coupled Hamiltonian with inertial parameter $\mathcal J=\{(i,j):1\le i<j\le N\}$8, yielding sitewise independent parallel updates and a stationary law $\mathcal J=\{(i,j):1\le i<j\le N\}$9 that is close in total variation to the target Gibbs measure $\varphi(x_1,\dots,x_n)$0 when $\varphi(x_1,\dots,x_n)$1 is sufficiently small. Here the sampling problem is controlled not by pair selection but by the trade-off between parallelism and approximation error (Pra et al., 2012).

Taken together, these results show that “pairwise interaction sampling” in stochastic-process literature refers to a different design axis from active comparison querying. The sampled object is the full configuration or state, but exactness and efficiency are achieved by exploiting locality, comparison-induced reversibility, local stability, or parallel product-form updates (Fotakis et al., 2022, Moka et al., 2019, Flint et al., 2022, Pra et al., 2012).

6. Covering feasible feature pairs in configurable software systems

In configurable software systems, pairwise interaction sampling becomes a minimum-coverage optimization problem. The system is described by a propositional formula $\varphi(x_1,\dots,x_n)$2 over Boolean features, valid configurations are satisfying assignments, and an interaction is a set of $\varphi(x_1,\dots,x_n)$3 concrete literals, with the practical focus on $\varphi(x_1,\dots,x_n)$4. A valid configuration covers an interaction if the interaction is a subset of the configuration, and the objective is to find a minimum-cardinality sample of valid configurations that covers every feasible pairwise interaction (Fekete et al., 7 Oct 2025).

The complexity landscape is unusually strong. For constant $\varphi(x_1,\dots,x_n)$5, the decision version lies in $\varphi(x_1,\dots,x_n)$6, via SAT-oracle queries that count feasible interactions and test whether at least $\varphi(x_1,\dots,x_n)$7 interactions can be covered with at most $\varphi(x_1,\dots,x_n)$8 configurations. At the same time, the problem is $\varphi(x_1,\dots,x_n)$9-hard, and the hardness extends from pairwise sampling to any constant $s_i$0 (Fekete et al., 7 Oct 2025). This rules out the interpretation of pairwise test generation as a routine greedy-cover instance; the presence of feature constraints changes the problem class materially.

The solver architecture called Sammy addresses the problem through sampling-safe preprocessing, heuristic construction, lower-bound computation, and exact SAT-based repair. The preprocessing pipeline includes failed literal elimination, equivalent literal elimination, bounded variable elimination on non-concrete features, clause vivification, subsumption removal, and universe reduction by implication under unit propagation. Lower bounds are obtained from cliques in a graph on feasible interactions, conservatively approximated by a computable subgraph $s_i$1 where an edge exists if

$s_i$2

Repair subproblems are encoded by making $s_i$3 copies of the feature model, introducing variables $s_i$4 that indicate whether interaction $s_i$5 is covered by copy $s_i$6, and enforcing coverage through clauses of the form $s_i$7 (Fekete et al., 7 Oct 2025).

The empirical results are notable for both optimality proofs and scale. On a benchmark set of 55 instances, Sammy solved $s_i$8 to provable optimality within one hour, compared with $s_i$9 for SampLNS. The largest publicly available Automotive02_V1–V4 instances contained between 301,619,972 and 535,654,597 feasible interactions before reduction, and these were solved to optimality after preprocessing and universe reduction (Fekete et al., 7 Oct 2025).

This software-engineering formulation is conceptually distinct from statistical pair selection, but it shares the same structural lesson as the learning and preference literatures: the pair is the correct primitive. Feasibility, lower bounds, and exact optimization all operate on interactions themselves, not on marginal features alone. A plausible implication is that pairwise interaction sampling should be regarded less as a single algorithmic technique than as a design principle: whenever the objective, loss, or coverage requirement is fundamentally pair-defined, methods that elevate pairs to first-class sampling units dominate methods that treat them as by-products of unary sampling (Fekete et al., 7 Oct 2025, Davy et al., 1 Jun 2026).