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Useful to Whom? The Easy-versus-Coverage Boundary Moves with the Learner

Yangze Liu, Xiao-Long Yin, Zhongyi Han

arXiv:2610.00221v2cs.LGcs.AI

TL;DR

The paper asks whether a sample's usefulness belongs to the data or depends on the learner, and tests this by comparing easy-first and coverage samples with selected subsets frozen. The crossover shifts with learner properties, so the preferred selection regime depends on the learner that will use the data.

  • Problem

    Data-selection practice often treats sample usefulness as learner-independent, although learners differ in what they can extract from the same sample.

    Source: §1 · p. 1

  • Method

    The study measures the budget crossover between easy-first and coverage samples, freezes their selected subsets, and changes only the learner that trains on them.

    Source: §1 · p. 2

  • Results

    On frozen subsets, doubling ResNet-18 width moves m0 from 57 to 85 samples per class, while the Vision Transformer favors coverage at every tested budget.

    Source: §1 · p. 2

  • Takeaways & Limitations

    A selection strategy's preferred budget regime must be read against the learner that will use the data.

    Source: §5 · p. 10

    The boundary is measured for two easy-first probes against Herding, and low-resolution runs use a fixed 90-epoch schedule.

    Source: §5 · p. 9

Abstract

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Is the same data equally useful to every learner? The question is whether the value of a training sample belongs to the sample or arises in its interaction with the learner. We test it with a selection experiment. Under a per-class budget, two kinds of samples compete for the slots: easy samples, which a short training run learns first, and coverage samples, which spread over the class in feature space. Easy samples win at small budgets and coverage samples at large ones, and the budget at which the winner flips, the crossover m0, can be measured. We ask whether m0 is set by the data or by the learner. To separate the two, we freeze the selected subsets and change only the learner that trains on them. On low-resolution ImageNet-100, doubling the width of a ResNet-18 moves the crossover from 57 to 85 samples per class on the same subsets. Enlarging the input grid from 32 to 64 pixels at fixed image information moves it from 85 to 57, and a single stride change reproduces or removes this shift. The grid shift shrinks as width grows. Under the native ImageNet-1k protocol, a 4x range of width moves the boundary by at most a few samples per class. On a Vision Transformer, coverage wins at every measured budget, even with easy subsets that work on the convolutional learner. The budget at which one kind of sample stops being the more useful kind therefore belongs to the learner, and the preferred regime of a selection strategy has to be read against the learner that will use the data.

1 INTRODUCTION

The paper asks whether sample usefulness belongs to the data or depends on the learner, and tests this by freezing selected subsets while changing the learner. The measured crossover moves across learners, so selection preferences must be interpreted relative to the learner that will use the data.

1 INTRODUCTION

  • The paper frames sample usefulness as an open question because learners differ in what they can extract from samples and in how they fit small subsets.If usefulness depends on the learner, a strategy validated on one model does not guarantee the same winner for another.Source: §1 · p. 1
  • The experiment uses easy-first samples learned early in a short run and coverage samples spread across class feature space; their winner changes with budget.The crossover m0 is the budget at which the preferred sample type flips; freezing subsets removes the changing scorer as a confound.Source: §1 · p. 2
  • On the same frozen subsets, doubling ResNet-18 width moves the crossover from 57 to 85 samples per class; a Vision Transformer favors coverage at every tested budget.The easy set that helps the convolutional learner is no more useful than a random subset on the Vision Transformer.Source: §1 · p. 2
  • Enlarging the input grid at fixed image information moves the crossover left, a stride change reproduces or removes the shift, and the shift shrinks as width grows.The feature-map size inside the convolutional learner is identified as the spatial carrier of this effect.Source: §1 · p. 2
  • Under the native ImageNet-1k protocol, a 15× parameter range moves the boundary by only a few samples per class, challenging a samples-per-parameter explanation.The introduction states that spatial processing, rather than learner scale, sets the boundary in this setting.Source: §1 · p. 2

2 RELATED WORK

Prior work supplies methods for selecting data and a measured easy-versus-coverage crossover, but primarily studies data variation under a fixed learner. This paper instead varies the learner while holding selected subsets fixed.

2 RELATED WORK

  • Data-selection work spans feature-space coverage, training-dynamics scores, optimization objectives, proxy models, and hybrids that combine coverage with difficulty.Examples include Welling (2009), Sener & Savarese (2018), Toneva et al. (2019), and Paul et al. (2021).Source: §2 · pp. 2–3
  • Liu et al. (2026) introduced the easy-versus-coverage crossover in absolute samples per class, the LFrac probe, and Herding as the coverage anchor; this paper adopts that framework unchanged.Their work varies the data under a fixed learner, whereas this paper varies the learner under fixed data and proposes no selection method.Source: §2 · p. 3
  • Data valuation and cross-architecture studies examine sample worth, score portability, or subset advantage across learners, rather than whether the winning sample type's budget boundary moves.The paper distinguishes its question from these lines of work by freezing subsets and manipulating the learner.Source: §2 · p. 3
  • Sorscher et al. (2022) studied an easy/hard flip line with the learner fixed; visual inspection suggested near-constant samples per parameter, a prediction tested by the paper's width sweeps.Source: §2 · p. 3

3 EXPERIMENTAL SETUP

The experiments measure the easy-first versus Herding boundary while varying learner properties and, in the attribution design, changing only the learner that trains on frozen subsets. The study combines low-resolution ImageNet-100 mechanism experiments with ImageNet-1k checks.

3.1 MEASURING THE CROSSOVER

  • LFrac keeps images with the highest mean correctness over early epochs, while EL2N keeps those with the lowest early-checkpoint softmax-to-onehot error norm.The two probes select substantially different images: their easy sets overlap by about 17%.Source: §3.1 · p. 3
  • The crossover m0 is the first interpolated positive-to-negative gap crossing; if the gap never changes sign in the measured range, the result is a censored bound.Selectors, data, and protocol remain fixed while the learner changes in the experiments.Source: §3.1 · p. 4

3.2 LEARNER KNOBS AND THE FROZEN-SUBSET DESIGN

  • The learner is varied along capacity, input grid, and family; the a@b notation fixes retained information at a pixels and delivers it on a b-pixel grid.Grid-only comparisons hold information fixed, while width changes channel and parameter counts.Source: §3.2 · p. 4
  • The frozen-subset design reuses byte-for-byte the ResNet-18-selected easy and coverage lists, making the training learner the only changing factor.The self-selected design instead gives each learner its own proxy-selected easy set and fills in the map around the frozen design.Source: §3.2 · p. 4

3.3 DATA AND PROTOCOL

  • Mechanism experiments use ImageNet-100 with a 3 × 3-stem ResNet, 90-epoch SGD, and a cosine schedule; full-scale checks use ImageNet-1k protocols.Load-bearing curves average three seeds, while the ImageNet-1k low-resolution ladder is single-seed.Source: §3.3 · p. 4

4 RESULTS

Across frozen subsets, changing the learner moves the easy-versus-coverage crossover; input grid and learner family also change which strategy wins. At native ImageNet-1k scale, width shifts the boundary only slightly, while low-resolution grid readings remain censored.

4.1 DOES THE BOUNDARY MOVE WITH THE LEARNER?

  • On frozen R18-selected subsets, widening ResNet-18 moves m0 from 57 to 85 samples per class; at 65, following the wrong verdict costs about 3 accuracy points.The boundary moves right because the coverage set loses more value on the wider learner, while the easy-first set loses no more than 2.5 points.Source: Table 1 · p. 5, §4.1 · p. 5

4.2 WHICH PART OF THE LEARNER MOVES IT? THE INPUT GRID

  • At fixed 32-pixel information, enlarging the input grid moves m0 from 85 to 57 samples per class, while raising information at a fixed grid moves it only to 53.The frozen-list transplant attributes about two thirds of the full grid shift to the learner and one third to the lists, with wide intervals.Source: §4.2 · p. 5
  • With the grid and frozen lists fixed, changing one stride moves m0 from 85 to 63 at layer 2 or 64 at layer 4; the reverse control returns it to 90.The layer-4 change affects only the map entering global average pooling, identifying internal feature-map size as the carrier of the grid effect.Source: §4.2 · pp. 5–6, Table 2 · p. 6, Figure 3 · p. 6

4.3 CAPACITY GATES THE GRID EFFECT

  • The grid shift is large at 0.5× and 1× width but collapses at 2×; at 64@64, m0 stays between 53 and 59 across widths despite a 16× parameter range.Equal compute increases move the boundary in opposite directions: grid doubling shifts it left by 28 samples at 1×, while width doubling at 32@64 shifts it right by 28.Source: §4.3 · pp. 6–7, Figure 4 · p. 7, Table 4 · p. 7

4.4 CHANGING THE LEARNER FAMILY REMOVES THE EASY-FIRST PHASE

  • On ViT-Tiny, coverage wins at every measured budget, and two independently selected easy sets reach only the uniform-sampling level.Both patch sizes and probes yield m0 < 25; the R18 easy set gains 0.1–0.9 points over uniform, while Herding leads by 1–3 points. The ViT reaches 99.8% training accuracy at 180 samples per class, yet test accuracy at 250 stays near 30%.Source: §4.4 · p. 7, Figure 5 · p. 8

4.5 DOES IT HOLD AT FULL SCALE?

  • Across the 4× native ImageNet-1k width range, LFrac’s boundary moves under one sample per class and EL2N’s about seven; all three low-resolution grid readings are censored.At 100 samples per class, the gap decreases strictly with grid size across the low-resolution cells; the crossover lies beyond the measured range.Source: §4.5 · pp. 8–9

5 DISCUSSION AND CONCLUSION

The crossover between easy-first and coverage selection depends on the learner: frozen subsets can favor different strategies for different learners. Within convolutional learners, pooled feature-map size tracks the boundary, while the width effect is conditional and remains unexplained.

5 DISCUSSION AND CONCLUSION

  • On frozen subsets, doubling ResNet-18 width moves the crossover from 57 to 85 samples per class; the preferred strategy therefore depends on the learner.The conclusion also reports that a Vision Transformer has no easy-first phase within the measured range.Source: §5 · p. 9
  • Across convolutional learners, m0 tracks the number of positions in the pooled feature map: 16-position learners read 85–90, while 64-position learners read 53–67.The authors offer spatial redundancy of prototypical easy samples as a hypothesis, and report that the prediction has so far held.Source: §5 · p. 9
  • Width adds channels rather than positions, and its effect on the boundary appears only with information-poor inputs; the proposed explanation remains open.The coverage set loses more accuracy than the easy set on the wider learner, while the same width increase moves nothing at 64@64 or under the native protocol.Source: §5 · p. 9
  • The conclusions are limited to two easy-first probes against Herding, and the low-resolution boundary positions reflect a fixed 90-epoch schedule.The 64@64 sweep and ImageNet-1k ladder use self-selected lists, while the ladder uses one seed; those results map directions rather than isolate causes.Source: §5 · pp. 9–10
  • A selection strategy’s preferred budget regime must be judged against the learner that will use the data, because the same budget can favor opposite strategies.The paper identifies pooled feature-map size as one learner property associated with the boundary, but not parameter count.Source: §5 · pp. 9–10

A TINYIMAGENET GRID-LADDER CORROBORATION

On TinyImageNet, the grid ladder supports the same directional ordering seen in the main results: denser learner grids bring coverage wins earlier. The 64@128 reading is corroborative rather than load-bearing.

A TINYIMAGENET GRID-LADDER CORROBORATION

B.1 SIX-CELL MAP AND TWO PROBES

The appendix tests width and grid effects with fixed selection features and probes crossover readings across several controls. Its results qualify the width pattern, localize part of the grid effect, and reject a single-scalar load rule.

B.1 SIX-CELL MAP AND TWO PROBES

  • LFrac and EL2N easy sets overlap by only approximately 16%–17%, yet yield the same capacity-gating verdict.Capacity raises full-data accuracy ceilings for 32@32 and 32@64 by 3.0 and 2.1 percentage points, respectively.Source: §B.1 · p. 13

B.2 DENSIFIED W2 BUDGETS AND TREATMENT OF SHALLOW CROSSINGS

  • For W2 in 32@64, m0 is 85 with bracket [80,90]; in 32@32, the shallow crossing is approximately 95 with bracket [80,110].Non-monotonic readings make the wider bracket more appropriate than automatic interpolation.Source: §B.2 · p. 13

B.3 CROSS-GRID SUBSET TRANSPLANT (2 × 2)

  • The cross-grid 2 × 2 readout estimates learner effects of 18.3/18.4 samples per class and list effects of 9.3/9.4, with interaction approximately 0.05.The bootstrap intervals support positive main effects, while the interaction interval spans zero; point-estimate additivity is not statistical equivalence.Source: §B.3 · p. 14

B.4 THE 64@64 WIDTH SWEEP

  • In 64@64, m0 is 57.1/53/59.1 samples per class at widths 0.5×/1×/2×, a narrow spread of approximately 6 despite a 5.5-point ceiling increase.The crossing brackets overlap and the shallow crossings do not establish statistical equivalence or a U-shaped width dependence.Source: §B.4 · p. 14

B.5 REJECTION OF THE PRE-REGISTERED LOAD LAW

  • The preregistered load law is rejected: equal-load conditions yield m0≈108 and 85, while the highest-load condition yields 64 rather than a value below 57.The paper therefore does not present a single formula for m0.Source: §B.5 · p. 14

B.6 STRIDE CONTROLS: RECOVERING AND REMOVING THE GRID EFFECT

Stride controls test whether the feature-map ladder carries the grid effect while holding the input grid fixed. The layer-4 control indicates that the pooled-map rung is sufficient for the measured boundary shift.

B.6 STRIDE CONTROLS: RECOVERING AND REMOVING THE GRID EFFECT

  • The layer-4 stride control gives m0=63.7 [55,65], within a point of the layer-2 control’s 62.7 and below the native 32@32 reading.Doubling only the map entering global average pooling, from 16 to 64 positions, moves the boundary by the full learner-side share; enlarging earlier rungs adds no measurable shift beyond it.Source: §B.6 · p. 15
  • The forward layer-2 control at grid 32 reads 62.7 versus 66.6 for the 32@64 learner on the same lists; the reverse stem control reads 89.8 versus 75.6 for the 32@32 learner.The controls move the boundary in the direction predicted by their feature-map ladders, though the matches are structural rather than exact.Source: §B.6 · p. 15
  • Whether an earlier rung alone would suffice while keeping the pooled map at 4 × 4 is not tested.The reported layer-4 control tests the last rung, not isolated earlier rungs.Source: §B.6 · p. 16

C.1 FOUR LEFT-CENSORED MEASUREMENTS

For both ViT patch sizes, easy-first never leads at any measured budget, and a convolutional learner’s easy subset does not restore its advantage when transplanted to ViT.

C.1 FOUR LEFT-CENSORED MEASUREMENTS

  • Across the four patch-size and selector configurations, easy-first remains close to Uniform, while the similar P8/P4 ceilings do not resolve whether patch granularity shifts the exact m0.The full-data ceilings are 49.14% for P8 and 49.39% for P4.Source: §C.1 · p. 16
  • All nine LFrac−Herding gaps for P8 and P4 are negative, so both crossovers are left-censored at m0 < 25.Local rebounds do not change the censoring verdict.Source: Figure C1 · p. 16, §C.1 · p. 16

C.2 TRANSPLANTING THE R18 EASY SET TO VIT

  • Transplanting the R18 easy set to ViT yields only +0.1 to +0.9 points over Uniform at the listed budgets, while Herding leads Uniform by approximately 1–3 points.The ViT and R18 easy sets overlap by approximately 5.6%, close to the 5.1% random baseline.Source: §C.2 · p. 17
  • The ViT crossover readings remain imprecise because accuracy at m=250 is approximately 28%–30%, well below the approximately 49% ceiling.Training accuracy reaches 99.8% at m=180, and whether another recipe or pretraining changes the censoring verdict is untested.Source: §C.2 · p. 17

D.1 LOW-RESOLUTION GRID LADDER AND TWO DATA REPLICAS

The ImageNet-1k low-resolution ladder supports an ordering of gap erosion across grids, but its single-seed results and tail behavior warrant censored bounds rather than precise crossover estimates.

D.1 LOW-RESOLUTION GRID LADDER AND TWO DATA REPLICAS

  • Across the two data replicas, the low-resolution ladder compares 32@32, 32@64, and 32@128 with 32px information, while constructing subsets independently because class-index orders differ.All three conditions use the 3 × 3-stem ResNet 32x32.Source: §D.1 · p. 17
  • Every absolute cross-seed difference at m=40/100 is below 2 percentage points, supporting faster erosion for 32@64 but only m0 > 250 for both 32@32 and 32@64.The second ModelScope seed gives 32@32/32@64 gaps of +9.48/+7.79 at m=100.Source: §D.1 · p. 17

D.2 SHARED BUDGETS AND TAIL FOR 32@128

  • The gap generally approaches zero as the grid becomes denser, but the 32@128 tail rebounds, so the ladder supports erosion-rate ordering and censored lower bounds rather than an interpolated flip.For 32@128, the gap rises from +2.28 at m=100 to +2.98/+2.73 at m=130/180; only m0 > 180 is reported.Source: §D.2 · pp. 17–18, Table D1 · p. 18, Table D2 · p. 18

D.3 CAPACITY SWEEP UNDER THE NATIVE-224PX PROTOCOL

Under the native-224px protocol, width changes the measured crossover only modestly, with paired-seed analyses distinguishing the EL2N and LFrac responses. The experiments compare widths on frozen subsets and use coarse exploratory uncertainty estimates.

D.3 CAPACITY SWEEP UNDER THE NATIVE-224PX PROTOCOL

  • Within each probe, widths train on the same frozen easy sets and fixed R18-Herding coverage set, isolating width as the change; increasing width raises both arms’ accuracy in near synchrony.At 64.1 samples/class, both arms increase by 2.5–4.5 percentage points with each width increase.Source: §D.3 · pp. 18–19
  • LFrac20−Herding m0 values are 43.8/42.5/44.4 for 0.5×/1×/2× widths, all within the shared measurement bracket [38.4,51.2].The figure plots per-budget gaps at three widths on the same frozen subsets.Source: §D.3 · p. 18, Figure D1 · p. 19
  • EL2N20−Herding m0 values are 31.7/34.0/38.4 for 0.5×/1×/2× widths; W2 is reported as approximately 38–40 because its flip lies at the bracket edge.All three share measurement bracket [25.6,38.4].Source: §D.3 · p. 19, Figure D1 · p. 19

D.4 PAIRED SEED UNCERTAINTY FOR FROZEN-SUBSET WIDTH CONTRASTS

  • Uncertainty uses 27 paired draws with replacement from three physical seeds, so the resulting percentile intervals are coarse exploratory estimates, not equivalence tests or independent experiments.Intervals for rows with unresolved crossings are conditional on both crossings being observed.Source: §D.4 · p. 19, Table D3 · p. 20
  • The native-protocol width response is smaller than the ImageNet-100 frozen-subset shift: EL2N shifts right in every bootstrap draw and physical seed, while the LFrac endpoint interval includes zero.The LFrac W1 missing crossing is left-censored rather than extrapolated.Source: §D.4 · p. 20

E.1 SELECTORS AND SHORT-TRAINING PROXIES

The short-training proxies define two learnability-based easy-sample selectors, while Herding supplies a fixed-feature coverage comparator. The LFrac and EL2N easy lists are substantially different measurements, not seed-level variants of one list.

E.1 SELECTORS AND SHORT-TRAINING PROXIES

  • LFrac selects within-class samples with the highest mean per-image correctness during early proxy-training epochs, whereas EL2N selects the lowest softmax–onehot L2 errors.ImageNet-100 uses 10-epoch proxies; the native-224px sweep uses a shared 20-epoch R18 proxy.Source: §E.1 · p. 20
  • LFrac and EL2N easy sets overlap by approximately 17% on ImageNet-100, above the approximately 5% random baseline but below LFrac’s approximately 37%–41% seed self-overlap.Their list differences therefore exceed the seed noise of one method.Source: §E.1 · p. 20
  • Herding greedily matches each class’s feature mean using a fixed 10-epoch R18 extractor, while Uniform samples uniformly within each class.The cross-selector experiment freezes R18 easy and coverage sets while changing only the downstream learner.Source: §E.1 · p. 20

E.2 LEARNERS AND TRAINING CONFIGURATIONS

The appendix specifies learner configurations, exact-budget handling, crossover reporting rules, and a run inventory. Comparisons keep training schedules fixed within controlled settings and avoid extrapolating unobserved crossovers.

E.2 LEARNERS AND TRAINING CONFIGURATIONS

  • Low-resolution ResNet runs use a 3×3 stem, while native-224px runs use a 7×7 stem; load-bearing ImageNet-100, ViT, and native-224px results use three paired proxy and downstream seeds.The low-resolution and native-224px settings are distinct stem families, and load-bearing comparisons remain within a family.Source: §E.2 · pp. 20–21

E.3 BN-SAFE BUDGETS AND EXACT SUBSET SIZES

  • Selected subsets are truncated only along the preselected within-class order to meet exact totals and keep batch-normalization remainders at least 16, preserving fixed features and the coverage criterion.Densified W2 budgets use exact totals of 9000/11000, with remainders of 40/248.Source: §E.3 · p. 21

E.4 MEASUREMENT STATUS OF m0

  • m0 is linearly interpolated only when adjacent measured budgets flip positive-to-negative; otherwise the paper reports one-sided censored bounds and avoids extrapolated or spurious precision.Measurement brackets encode budget discretization, not statistical confidence intervals.Source: §E.4 · p. 21, Table E1 · p. 22

F RUN INVENTORY

  • The run inventory covers 1,839 result files, and main-text experiment numbers were re-aggregated and checked against reference values with zero verification failures.Data-side results quoted from Liu et al. (2026) are cited rather than re-verified in the inventory.Source: F RUN INVENTORY · p. 22

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