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Counterfactual Constraint-Conditioned On-Policy Distillation for Multi-Constraint Instruction Following

Yanzhao Zheng, Yuanqiang Yu, Tianze Xu, Chao Ma, Zhentao Zhang, Jihuai Zhu, Baohua Dong, Hangcheng Zhu, Ruohui Huang

arXiv:2609.27421v1cs.LGstat.ML

TL;DR

Multi-constraint instruction following remains challenging, while existing verifier-based RL and full-context OPD do not directly extract per-constraint information from the teacher. CC-OPD uses leave-one-out counterfactual teacher scoring to shape token-level OPD rewards, and achieves the highest average among evaluated methods across two Qwen settings and seven benchmarks.

  • Problem

    Even strong instruction-tuned models routinely violate some simultaneous requirements, while verifier-based RL and full-context OPD leave per-constraint teacher information unused.

    Source: §1 · p. 1

  • Method

    CC-OPD scores the same student rollout with the frozen teacher under the full instruction and each leave-one-out context, then adds summed, clipped per-token shifts to vanilla OPD reward.

    Source: §1 · p. 2, §3 · p. 3

  • Results

    Across two Qwen settings and seven benchmarks, CC-OPD achieves the highest average among evaluated student-training methods; on MulDimIF, a 1.5B student beats its own 7B RL-trained teacher by +2.8 pp overall and +6.3 pp on the hardest split.

    Source: §1 · p. 2, §4.2 · p. 5

  • Takeaways & Limitations

    The results support an ablative supervision approach that queries the same frozen teacher across counterfactual constraint subsets to expose per-constraint signals.

    Source: §6 · p. 9

    The construction requires a white-box teacher with per-token log-probability access under multiple prompts; generation-only APIs are outside its current scope.

    Source: Limitations · p. 9

Abstract

from arXiv · show

Multi-constraint instruction following requires a model to respond to a query under many simultaneously active constraints. Even strong instruction-tuned models still routinely violate some of them. Existing approaches either augment supervision with sequence- or token-level RL rewards from external verifiers or learned graders, or use on-policy distillation (OPD) against a single full-context teacher whose probability mass becomes diluted as more constraints become simultaneously active. We propose CC-OPD (Counterfactual Constraint-Conditioned On-Policy Distillation), which inverts the standard supervision-generation direction in distillation. Rather than enriching the teacher with information beyond what the student sees, CC-OPD ablates each constraint from the teacher's conditioning in turn, and constructs the per-constraint signal from the resulting per-token probability differentials. The resulting per-token leave-one-out log-likelihood shifts are summed, clipped, and added to the vanilla OPD reward as a token-level shaping term. All shaping terms are obtained from the frozen teacher, without an external verifier during distillation, and the reward equals vanilla OPD wherever the aggregate shift is zero. Across two Qwen model pairs and seven benchmarks, CC-OPD achieves the highest average among all evaluated student-training methods. A 1.5B student trained with CC-OPD surpasses its own 7B RL-trained teacher on the MulDimIF benchmark.

1 Introduction

Multi-constraint instruction following remains difficult, and existing reward-based RL and full-context OPD leave per-constraint teacher information unused. CC-OPD extracts that information through counterfactual teacher scoring and leads the evaluated methods across benchmarks.

1 Introduction

  • As prompts carry more simultaneous constraints, even strong instruction-tuned models routinely violate some, and the gap grows with constraint difficulty and number.Multi-constraint prompts can specify format, length, style, content, safety, or citation requirements.Source: §1 · p. 1
  • Rubric-based RL uses external graders, while full-context OPD conditions on all requirements jointly; neither extracts per-constraint information directly from the teacher.The paper calls the resulting difficulty of disentangling an individual constraint's contribution constraint dilution.Source: §1 · p. 1
  • A 1.5B CC-OPD student beats its own 7B RL-trained teacher by +2.8 pp on MulDimIF overall and +6.3 pp on the hardest split; CC-OPD leads average across seven benchmarks.The training trajectory crosses the teacher's accuracy ceiling on both overall score and the hardest Level-4 split.Source: Figure 1 · p. 2, §1 · p. 2
  • CC-OPD scores the same student response under the full instruction and each leave-one-out teacher context, then adds summed, clipped per-token shifts to vanilla OPD reward.The student remains conditioned on the full instruction, and a zero aggregate shift leaves the vanilla reward unchanged.Source: §1 · p. 2
  • The evaluation compares CC-OPD with SFT, verifier-based RL, OPD variants, and prompt-augmentation methods, alongside counterfactual-design and efficiency analyses.The reported benchmark scope includes two Qwen settings and seven instruction-following benchmarks.Source: §1 · p. 2

2 Preliminaries

Multi-constraint instruction following asks a model to satisfy several natural-language requirements in one response. Sampled-token OPD replaces full per-token distribution matching with a single-token estimate used as a PPO reward.

2.1 Multi-Constraint Instruction Following

  • A response must satisfy a set of natural-language constraints, commonly covering requirements such as length, formatting, taboos, tone, or citation format.The query and constraints are combined in one instruction, and realistic samples often contain several active requirements.Source: §2.1 · p. 2

2.2 On-Policy Distillation with Sampled-Token PPO

  • Canonical OPD distills a frozen teacher into a student using token-level divergence on sequences sampled from the student itself.The student rollout is conditioned on the query and full constraint set.Source: §2.2 · p. 2
  • Because full per-token vocabulary distributions are memory-bound at large vocabulary sizes, sampled-token PPO estimates the KL using the realized student token and treats its negative as reward.This requires only one teacher and one student log-probability per token; appropriately tuned sampled-token OPD can match top-k OPD end-task performance.Source: §2.2 · p. 3

3 CC-OPD

CC-OPD addresses constraint dilution by recovering a per-token signal for each constraint from counterfactual teacher scoring on the same student response. It sums and clips those shifts before adding them to the vanilla OPD reward.

3 CC-OPD

  • CC-OPD restores per-constraint resolution by querying the frozen teacher under the full instruction and each leave-one-out context while keeping the student conditioned on the full constraint set.The method uses the same student rollout across teacher contexts, varying only the prompt.Source: §3 · p. 3

3.1 Per-Constraint Counterfactual Teacher Scoring

  • For each constraint, the teacher forced-scores the same student response with that constraint removed, and the leave-one-out log-likelihood shift gives a per-token contrast.A positive shift means the constraint increases the teacher's probability for the realized token; near-zero means it is irrelevant, while negative shifts can indicate suppression or an unrelated prior shift.Source: §3.1 · pp. 3–4
  • Leave-one-out contrasts retain the effects of other active constraints, unlike isolated single-constraint scoring, which discards inter-constraint interactions.The same rollout is shared across the teacher passes, so the contrast isolates the marginal effect without additional rollouts.Source: §3.1 · p. 4

3.2 Counterfactual Reward Shaping

  • CC-OPD sums per-constraint shifts, symmetrically clips the aggregate, and adds the shaping term to vanilla OPD reward for the PPO update.Clipping bounds the shaping contribution; when the aggregate shift is zero, the reward reduces exactly to vanilla sampled-token OPD.Source: §3.2 · p. 4

4 Experiments

Across two Qwen model pairs, CC-OPD achieves the highest average among evaluated student-training methods on seven instruction-following benchmarks. It also surpasses teacher performance on MulDimIF and improves training efficiency and counterfactual-variant performance in the reported comparisons.

4.1 Experimental Setup

  • Experiments use a GRPO-IR-tuned Qwen2.5-7B teacher with a Qwen2.5-1.5B student and a same-size Qwen3-4B student–teacher pair.The comparison includes SFT, rubric-based RL, vanilla OPD, OPD variants, and other distillation methods; CC-OPD derives rewards from teacher log-probabilities without an external grader.Source: §4.1 · pp. 4–5
  • Evaluation covers seven instruction-following benchmarks, with instruction-level accuracy defined as the percentage of responses satisfying every prompt constraint.CC-OPD uses λ = 2.0 and c = 5.0; IFEval, IFBench, and MulDimIF select these values, while four other benchmarks serve as transfer checks.Source: §4.1 · p. 5

4.2 Main Results

  • 51.4% on Qwen2.5 and 64.3% on Qwen3 are CC-OPD’s best seven-benchmark averages, leading the strongest baselines by 2.5 pp and 1.5 pp, respectively.It leads on six of seven individual benchmarks for Qwen2.5 and is best or tied on five of seven for Qwen3.Source: §4.2 · p. 5, Table 1 · p. 6

4.3 Student Surpasses Teacher

  • 72.9% on MulDimIF overall and 59.4% on Level-4 put the Qwen2.5 1.5B CC-OPD student 2.8 pp and 6.3 pp above its 7B teacher.On Qwen3, the student exceeds its same-size teacher on five of seven benchmarks; CC-OPD has the largest MulDimIF margin among tested OPD variants in both pairs.Source: §4.3 · pp. 5–7

4.4 Gradient-Step Efficiency

  • At isoaccuracy, CC-OPD uses about 30% of OPD’s total FLOPs on Qwen2.5 and 53% on Qwen3, despite 4.63× per-step teacher overhead on Qwen2.5.On Qwen2.5, it matches OPD’s smoothed MulDimIF peak in 15.2× fewer gradient steps overall and 16.6× fewer on Level-4; OPD does not reach CC-OPD’s eventual peak within 48,000 steps.Source: §4.4 · p. 7

4.5 Counterfactual Variant Ablation

  • 51.4% is the leave-one-out variant’s strongest seven-benchmark average, 0.9 pp above LA and RSS and 2.7 pp above NAO.LOO outperforms LA on six of seven benchmarks; Table 2 compares four counterfactual variants under identical training hyperparameters.Source: §4.5 · p. 7, Table 2 · p. 8
  • LOO weights an order-k interaction by k, whereas LA weights every non-empty interaction uniformly; NAO is 2.7 pp lower on average and 6.5 pp lower on MulDimIF.LA remains a more efficient approximation when teacher inference dominates: it uses two teacher passes per sample, versus |C| + 1 for LOO and RSS.Source: §4.5 · p. 8
  • LOO improves accuracy over LA on all four MulDimIF tiers: +2.3, +0.6, +2.2, and +2.3 pp on L1–L4.The gains are positive but not monotonic with difficulty; the LOO–LA difference depends on signed higher-order terms and reward clipping, not constraint count alone.Source: Table 3 · p. 8, §4.5 · p. 8

5 Related Work

Related work uses rubric-based reinforcement learning to turn constraints into externally graded rewards, while on-policy distillation matches a teacher on student rollouts. CC-OPD instead uses constraint-wise leave-one-out teacher scoring.

On-policy distillation

Rubric-based RL for instruction following

Contrastive likelihood signals

6 Conclusion

The paper presents CC-OPD, which obtains per-constraint signals by leave-one-out scoring of student rollouts with a frozen teacher. It reports the highest average among evaluated methods across two Qwen settings and the largest student-over-teacher MulDimIF margin among tested OPD variants.

6 Conclusion

  • CC-OPD obtains per-constraint signals through a frozen teacher’s leave-one-out forced scoring on the same student rollout.Source: §6 · p. 9
  • Across two Qwen settings and seven benchmarks, CC-OPD has the highest average among evaluated student-training methods and the largest student-over-teacher MulDimIF margin among tested OPD variants.The paper frames this as ablative teacher conditioning, rather than supervision enriched with external rubrics or graders.Source: §6 · p. 9

Limitations

CC-OPD is evaluated on a narrow set of models, data, and objectives, and its counterfactual scoring has practical and prompt-format requirements. The appendix specifies the training setup and HIR-16K corpus used in these experiments.

Limitations

  • Generalization beyond HIR-16K and two Qwen model pairs—including to other corpora, model families, multi-turn use, or system-prompt-steerable settings—remains untested.The evaluated pairs are Qwen2.5-1.5B/7B and Qwen3-4B/4B.Source: Limitations · p. 9
  • The method requires a white-box teacher with per-token log-probability access under multiple counterfactual prompts, excluding generation-only APIs.Each sample also requires |C| + 1 teacher forward passes, adding per-step compute, memory use, and latency; total FLOPs were lower at matched accuracy on the two evaluated pairs.Source: Limitations · pp. 9–10
  • Experiments use only the sampled-token reverse-KL OPD objective, so effectiveness with other OPD objectives is untested.Source: Limitations · p. 9
  • Counterfactual construction assumes constraints can be removed by byte-aligned span deletion from inline natural-language instructions; structured fields or programmatic templates may need another construction.Source: Limitations · p. 9

A.1 Training Hyperparameters for OPD-Family Methods

  • CC-OPD is trained for 1,500 verl-steps, while OPD baselines run for 6,000 and are reported at their step-3,000 saturated checkpoints.The stated horizon reflects validation saturation: CC-OPD plateaus by step 1,500, while baselines plateau by step 3,000.Source: §A.1 · p. 12

A.2 HIR-16K Training Corpus

  • HIR-16K contains 16,968 multi-constraint samples with 5–17 constraints each (mean 7.25), spanning rule-based, table-structured, and LLM-judged categories.Source: §A.2 · p. 12

A.3 Teacher Training and GRPO Baselines

This section describes the teacher and baseline recipes, counterfactual variants, evaluation protocol, and analyses of how LOO and LA weight constraint interactions. The comparisons show that LOO often ranks highest among the tested counterfactual variants, while its signal differs from LA in scale and direction.

A.3 Teacher Training and GRPO Baselines

  • The GRPO-IR and GRPO-CR baselines differ in reward aggregation: instruction-level reward requires every constraint to pass, while constraint-level reward averages satisfaction across constraints.Source: §A.3 · p. 12
  • Self-CC-OPD averages 61.9 versus DR-CFG’s 61.0 on Qwen3-4B, while on Qwen2.5-1.5B their averages are 37.7 and 37.5, respectively.The per-benchmark comparisons vary by model pair; on Qwen3-4B self-CC-OPD ties FollowBench and is 0.1 pp lower on CFBench.Source: §A.3 · p. 14

A.4 OPD Data-Augmentation Baselines

  • OPD augmentation adds N rewritten rows per example by removing one constraint, retaining random complements, or keeping one constraint; the enlarged corpus is roughly eight times larger.These variants use vanilla sampled-token OPD without counterfactual reward shaping.Source: §A.4 · p. 14

B.3 LA and LOO sum as different weighted aggregations

  • LA weights each non-empty Möbius interaction by 1, whereas LOO weights it by its order |T|; a nonzero residual certifies higher-order interactions, but cancellation can yield zero.Source: §B.3 · p. 16
  • RSS has expected interaction weights that rise with order but with diminishing increments, and its unnormalized sum changes pre-clip reward scale and adds subset-sampling variance; NAO retains only first-order effects.Source: §B.3 · p. 16

B.4 Empirical validation: the additivity hypothesis is violated

  • For these measured signals, the LOO–LA residual is 0.918, signed Pearson correlation is 0.665, sign disagreement is 31.9%, and LOO’s aggregate signal norm is 13.7% larger.The nonzero residual rejects strict additivity for these measured signals and quantifies an aggregate weighted residual, not the magnitudes of individual Möbius coefficients.Source: §B.4 · p. 17

B.5 What LOO retains beyond LA: the K×T sensitivity map

  • LOO’s per-constraint sensitivity map is concentrated: one dominant constraint supplies 42.6% of per-token signal mass on average, versus 15.0% under uniform attribution.Estimated active constraints per token are 4.46 versus a nominal mean of 6.63; LA alone cannot compute these per-constraint map statistics.Source: §B.5 · p. 17

B.6 Cross-model comparison of counterfactual variants

  • LOO scores highest in three of four evaluated cells for each model pair—IFEval, MulDimIF Overall, and Level-4—while RSS scores highest on IFBench.Source: §B.6 · p. 17

B.7 Theory–empirics bridge and limitations

The appendix contrasts LOO and LA at the reward-geometry level, gives an idealized interpretation of the shaping reward, and illustrates how per-constraint signals differ from aggregate signals in token-level cases. The three cases show LOO localizing constraint-relevant tokens or regions, including a token tied to a violated constraint.

B.7 Theory–empirics bridge and limitations

  • LOO and LA differ sharply in reward geometry, but their Avg gap is +0.9 pp alongside a 91.8% relative L1 residual and 31.9% token-level sign disagreement.Reward geometry does not uniquely determine the policy reached by stochastic, clipped PPO, and the policy-level gap remains an empirical outcome of optimization and evaluation.Source: §B.7 · pp. 17–18

B.8 Idealized implicit target distribution

  • The idealized full-expectation objective yields a unique exponential-tilt, product-of-experts-style target driven by teacher-side finite differences; experiments instead use sampled-token rewards with ratio-clipped PPO.The unclipped LOO sum cardinality-weights interaction coefficients, unlike LA's unit weighting; the full Möbius expansion has 2^N terms and is intractable beyond very small constraint sets.Source: §B.7 · p. 17, §B.8 · pp. 18–19

C Token-Level Reward Heatmap Case Studies

  • Three selected cases visualize per-constraint token shifts alongside aggregated LOO and LA signals, with color indicating shift direction and intensity indicating magnitude.The cases illustrate distinct LOO–LA disagreement modes; the individual examples were selected from a filtered pool.Source: C Token-Level Reward Heatmap Case… · p. 19

C.1 Case 1: Literal keyword and format-token alignment

  • LOO peaks on literal keyword and format tokens, whereas LA's top-15 excludes all four constraint-relevant tokens and highlights generic discourse markers instead.The LOO peaks include +11.25 and +8.57 on keyword tokens and +19.99 on the colon-newline token.Source: §C.1 · p. 20

C.2 Case 2: LA collapses to structurally meaningless tokens

  • For the postscript constraint, LOO peaks on “Post” and “script,” while LA's strongest red token is the end-of-message marker.LOO concentrates in the P.S. region; LA's signal also highlights a bullet marker and generic words across the response.Source: §C.2 · p. 21

C.3 Case 3: Constraint identity at a shared salient token

  • When the response lacks a required table, LOO assigns the largest absolute shift to “These” and identifies the table constraint, while LA supplies only a full-versus-empty aggregate.Both aggregate signals are strong at that token, but only the per-constraint LOO view retains constraint identity.Source: §C.3 · p. 23

C.4 Synthesis, selection methodology, and caveats

The case studies portray LOO as localizing literal constraint tokens, structural regions, and violations in ways LA cannot replicate. They were selected for strong divergence, so the examples do not represent the full sample distribution.

C.4 Synthesis, selection methodology, and caveats

  • Across three modes, LOO localizes literal satisfaction tokens, a structural response region, and a violation region that LA cannot identify from its aggregate signal.The cases illustrate distinct functions of the K × T attribution matrix: literal keyword peaks, regional concentration, and violation-region marking.Source: §C.4 · p. 25
  • The three examples were the highest-scoring representatives of distinct disagreement modes, selected from a filtered pool of 169 samples and 80 inspected candidates.The filter required top-20 token Jaccard ≤0.30, sign disagreement ≥35%, and residual L1 at least equal to the LA signal's L1 magnitude.Source: §C.4 · p. 25
  • The visualizations describe reward geometry, not PPO trajectories or final policies, and the divergence filter over-represents high-disagreement cases.On samples where one constraint dominates full-context conditioning, LA and LOO sum partially agree; full-sample statistics provide the non-cherrypicked summary.Source: §C.4 · p. 25

D Aggregation Method Ablation

The ablation compares sum, mean, and top-k aggregation of per-constraint shifts. Sum leads the tested variants on MulDimIF Overall for both model pairs, while Level-4 differences are narrower.

D Aggregation Method Ablation

  • Mean rescales each sample's unclipped shaping signal by |C|, but its empirical comparison was not run because a fair test would require retuning λ and potentially the clip threshold.The reported empirical ablation covers sum and the two top-k variants.Source: D Aggregation Method Ablation · p. 25
  • Top-k discards all but the k largest-magnitude shifts at each token; its lower Overall scores suggest lost multi-constraint interaction contributions outweigh any credit-assignment gain.Sum also avoids per-token argmax computation and incorporates all |C| counterfactual teacher passes.Source: D Aggregation Method Ablation · pp. 25–26
  • On Level-4, Top-k=2 leads Sum by +0.3 on Qwen2.5, while Sum ties Top-k=1 at 63.8 on Qwen3.The passage reports narrower aggregation differences on the harder split.Source: D Aggregation Method Ablation · p. 25
  • Qwen2.5: Sum is +1.2 over Top-k=2 and +1.9 over Top-k=1; Qwen3: Sum is +2.5 and +1.8, respectively, on MulDimIF Overall.Table 8 compares MulDimIF accuracy under the three aggregation rules with other hyperparameters fixed.Source: D Aggregation Method Ablation · p. 25, Table 8 · p. 26

E Sensitivity to Shaping Strength and Clip Threshold

The sweep tests how shaping strength and clipping affect CC-OPD performance. The paper-main setting is the only tested joint optimum across all three benchmarks, while the separate parameter sweeps identify λ=2 and c=5 as best among tested values.

E Sensitivity to Shaping Strength and Clip Threshold

  • Clipping follows summation, so one extreme contrast or several moderate contrasts can produce a shift capped at |δ̂(t)|≤c.The absolute shaping contribution is bounded by λc per token.Source: E Sensitivity to Shaping Strength and… · p. 26
  • With c=5, all three benchmark scores peak at λ=2 among tested values; IFBench drops 2.0 pp at λ=4.IFEval scores 73.7, 75.4, 76.0, and 75.7 across λ={0.5, 1, 2, 4}; MulDimIF scores 71.1, 71.4, 72.9, and 70.9.Source: E Sensitivity to Shaping Strength and… · p. 26
  • At λ=2, c=5 gives the highest score on all three benchmarks among five tested thresholds, with non-monotonic performance across c.The score ranges across c are 2.4 pp on IFEval, 2.6 pp on IFBench, and 2.3 pp on MulDimIF Overall.Source: E Sensitivity to Shaping Strength and… · p. 26
  • The (λ=2.0, c=5.0) paper-main cell is the only tested setting best on all three benchmarks, scoring 76.0/28.2/72.9 on IFEval/IFBench/MulDimIF; every grid cell beats OPD.Even the extreme corners retain positive net gains, and settings within the tested neighborhood show comparable behavior without retuning.Source: E Sensitivity to Shaping Strength and… · pp. 26–27, Table 9 · p. 27

F Constraint Subsampling: How Many Counterfactual Contexts Are Needed?

The subsampling sweep finds that sparse leave-one-out scoring improves over OPD, but the value of denser attribution varies by benchmark. Higher ratios help more on constraint-stressed measures, while aggregate measures can saturate at lower ratios and use less per-step compute.

F Constraint Subsampling: How Many Counterfactual Contexts Are Needed?

  • Subsampling retains ⌈ρ|C|⌉ leave-one-out contexts and rescales their summed signal by |C|/⌈ρ|C|⌉ to preserve expected magnitude.The sweep holds λ=2.0 and c=5.0 fixed; per-step compute grows with ρ, and ρ=1.00 costs roughly 3× more per step than ρ=0.10.Source: F Constraint Subsampling: How Many… · p. 27
  • Table 10 compares selected checkpoints rather than equal update counts: ρ<1.0 runs for 3,000 steps, while ρ=1.00 runs for 1,500.Figure 5 reports full trajectories against both gradient steps and cumulative FLOPs.Source: F Constraint Subsampling: How Many… · p. 27, Figure 5 · p. 28
  • Every tested ρ>0 variant beats OPD on all four benchmarks; even ρ=0.10 lifts scores by +1.8 to +4.4 pp.The sparsest setting uses less than a full set of leave-one-out contexts per sample on average.Source: F Constraint Subsampling: How Many… · p. 27, Table 10 · p. 28
  • From ρ=0.10 to 1.00, IFBench rises 25.6→28.2 (+2.6 pp) and MulDimIF Level-4 rises 57.3→59.4 (+2.1 pp), while MulDimIF Overall saturates early.The Level-4 progression includes a minor non-monotone dip at ρ=0.50.Source: F Constraint Subsampling: How Many… · pp. 27–28, Table 10 · p. 28
  • On MulDimIF Overall, ρ=0.25 scores 72.5 versus 72.9 at ρ=1.00, while ρ=0.50 regresses to 71.6; IFEval ties at 76.0 for ρ=0.50 and 1.00.The reported saturation is attributed to easier sub-levels and the student’s general capability ceiling.Source: F Constraint Subsampling: How Many… · p. 27
  • No CC-OPD configuration is Pareto-dominated by OPD on either the gradient-step or FLOPs view; among CC-OPD settings, compute trades off against quality on constraint-stressed slices.At equal FLOPs, lower ρ has executed more steps; at equal steps, higher ρ uses more compute.Source: F Constraint Subsampling: How Many… · p. 28, Figure 5 · p. 28
  • The denser signal matters most where constraint-level discrimination is stressed, while ρ=0.25–0.50 recovers most aggregate gains at a fraction of per-step cost.The paper uses ρ=1.00 to characterize the upper end of leave-one-out estimator quality.Source: F Constraint Subsampling: How Many… · pp. 28–29

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