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Seeing Through Distractions: Context-Stable Attribution via the Least Core
Sai Ganesh Nagarajan, Toshinori Yamauchi, Hiroshi Kera
TL;DR
Averaging-based feature attributions can miss contextual cues whose effect differs between the complete genuine image and partial contexts. The paper formalizes these cues and compares semivalues with the stability-based least core, finding that attribution methods often rank distractors higher than the least core. Its results position the least core as a complementary attribution perspective, while noting computational limits.
Problem
A feature can reduce the target-class output in the complete genuine image while having a different marginal effect in partial contexts, complicating attribution based on averaging across contexts.
Source: §3.1 · p. 6Method
The paper formalizes contextual distractors and derives semivalue–least-core gap bounds under weak average marginal monotonicity, including conditions tied to sampling smaller contexts.
Source: §1 · p. 2Results
In controlled image-classification experiments and Salient ImageNet, Shapley and other attribution methods often assign distractors greater importance than the least core, even when genuine image content suffices for correct classification.
Source: §1 · p. 2Takeaways & Limitations
The least core offers a complementary perspective to the Shapley value by emphasizing coalitional stability rather than average marginal contributions.
Source: §5 · p. 11The least core has exponentially many coalition constraints, and efficient separation oracles may be difficult to obtain in practice.
Source: §5 · pp. 10–11
Abstract
from arXiv · showhide
Shapley-value-based explanations are widely used for feature attribution in machine learning. However, in computer vision, a feature may reduce the target-class score when added to the complete genuine image, even when the predicted label remains unchanged. We formalize such features as contextual distractors and investigate their attributions under semivalues and the least core. Under a weak average marginal monotonicity assumption, we derive quantitative semivalue--least-core gap bounds and sufficient conditions based on the probability of sampling smaller contexts. We further show that shifting semivalue weights toward larger contexts decreases the signed gap. Finally, controlled image-classification experiments and evaluations on Salient ImageNet provide strong empirical support for our theoretical findings.
1 Introduction
The introduction identifies a gap in averaging-based explanations for contextual cues and formalizes contextual distractors. It presents theory and experiments comparing semivalues with the least core.
1 Introduction
- A contextual distractor reduces the target-class score when added to the complete genuine image, even if the predicted label remains unchanged.Source: §1 · p. 2
- The paper derives semivalue–least-core gap bounds under weak average marginal monotonicity and characterizes how the gap changes as semivalues weight larger contexts more heavily.Source: §1 · p. 2
- Across controlled classification experiments and Salient ImageNet, Shapley and other attribution methods often rank distractors above the least core, even when genuine image content suffices for correct classification.Source: §1 · p. 2
- The paper distinguishes its focus on stability-based attribution from prior Shapley-style work that remains averaging-based [LL17] [SN20] [KZ22] [WJ23] [CKL23].Source: §1 · p. 3
- The analysis applies to every allocation in the least-core set, rather than depending on a particular least-core selector.Source: §1 · p. 3
2 Game Theory Preliminaries
The preliminaries frame feature attribution as allocating a game's value among players. They introduce semivalues as context-weighted averages and the least core as a stability-based allocation with the smallest feasible relaxation.
2 Game Theory Preliminaries
- In a transferable-utility game, players form coalitions with scalar values, and explanation methods allocate coalition value among features.Source: §2 · p. 3
2.1 Semivalues
- Symmetric semivalues generalize Shapley and Banzhaf values by assigning equal weights within each coalition size, with weights varying across sizes.Source: §2.1 · p. 4
- Group semivalues treat a feature group as one player; their weights induce sampling over context sizes and uniformly sampled contexts of each size.Source: §2.1 · p. 4
2.2 Least Core
- The least core relaxes coalition stability by the smallest ε that permits an efficient allocation, and the paper's theoretical results hold for every allocation in that set.Source: §2.2 · pp. 4–5
3 Explanations in Classifications with Contextual Distractors
The section models image explanations as cooperative games over masked features and studies how a feature’s target-class effect changes with image context. It formalizes contextual distractors and derives conditions for gaps between semivalue and least-core attributions, including how context weighting affects those gaps.
3 Explanations in Classifications with Contextual Distractors
- A fixed image becomes a game over pixel or feature coalitions, with each coalition evaluated by the target-class logit of its masked image.The target class remains fixed across coalitions, and theoretical results apply to any fixed target class.Source: §3 · pp. 5–6
3.1 Contextual Distractors
- A contextual distractor lowers the target-class output in the complete genuine-image context, while its contribution in smaller contexts may differ or even be helpful.The definition imposes no restriction on smaller-context contributions; average marginal monotonicity requires only that the average contribution not increase with context size.Source: §3.1 · p. 6
- When a distractor is helpful on average in smaller contexts but harmful in the complete genuine context, sufficiently little weight on larger contexts can make its semivalue attribution exceed its least-core attribution.The sufficient gap condition becomes easier as the contrast β + γ increases and harder as the least-core value ε⋆ increases.Source: §3.1 · pp. 6–7
- Under average marginal monotonicity, shifting semivalue weight stochastically toward larger contexts decreases the signed semivalue–least-core gap.The results apply to Shapley, weighted Banzhaf, and Beta-Shapley; the gap is nonincreasing in weighted Banzhaf parameter p and nondecreasing in a and nonincreasing in b for Beta-Shapley.Source: §3.1 · pp. 7–8
- Extreme context weighting need not improve informativeness: large-context emphasis can miss genuine support in smaller contexts, while small-context emphasis can miss harm in the complete image.The least core considers all coalitions through stability constraints rather than assigning fixed weights to context sizes.Source: §3.1 · p. 8
4 Experiments
Controlled marker and natural-image experiments test whether contextual distractors produce the predicted Shapley–least-core separation. Across both settings, distractors help in weak contexts but can suppress target scores in strong contexts, while Shapley assigns them more credit than the least core in the reported examples.
4.1 Controlled Marker-bias experiments
- Shapley assigns larger attribution than the least core to synthetic markers on most of 100 cat-class images.The marker's marginal effect decreases with stronger context and changes from positive to negative.Source: §4.1 · p. 8, Figure 2 · p. 9
4.2 Contextual Distractors in Natural Images
- Salient ImageNet analysis selects contextual-distractor regions by sign reversal across weak and strong contexts, requiring the expected sign in at least 60% of sampled contexts in each group.The analysis includes 354 candidates for ResNet-50 and 108 for ViT-B/16, restricted to images where predictions match ground-truth labels.Source: §4.2 · p. 9
- The natural-image comparison uses total distractor attribution and mask-weighted attribution, alongside prediction-active random-SAM control regions.Controls are generated as 200 regions per architecture.Source: §4.2 · p. 9
- Natural-image examples show fire and a U.S. flag changing from helpful in weak contexts to harmful in strong contexts.For the stove image, adding fire changes the target logit from 9.00 to 7.80 (∆D = −1.19); for the flagpole image, adding the flag changes it from 9.87 to 7.05 (∆D = −2.82).Source: §4.2 · p. 10, Figure 3 · p. 10
- For the stove image, Shapley assigns the fire AD = 3.60 versus Core's 0.85; for the flagpole image, it assigns the flag AD = 3.76 versus Core's 1.77.Core places relatively greater emphasis on the stove structure and pole, respectively.Source: §4.2 · p. 10
5 Limitations and Discussion
The least core is presented as a complementary perspective to Shapley, emphasizing coalitional stability rather than average marginal contributions. Its practical limitation is that solving the defining linear program may require difficult-to-obtain separation oracles.
5 Limitations and Discussion
- The least-core linear program involves exponentially many coalition constraints, limiting practical computation.Polynomial-time solution via the ellipsoid method requires an efficient separation oracle, which may be difficult to obtain in practice.Source: §5 · p. 10
- The least core complements Shapley by emphasizing coalitional stability rather than average marginal contributions.The paper frames it as a complementary attribution method for computer vision, where a large semivalue–least-core gap may indicate shortcut features or contextual distractors.Source: §5 · pp. 10–11
A.1 Proof of Theorem 2
- Theorem 2's proof establishes a general gap bound for any least-core allocation by combining efficiency with the least-core constraint.The proof then derives a sufficient condition using average marginal monotonicity and a contextual-distractor effect.Source: §A.1 · p. 14
A.2 Proof of Lemma 1
- The Lemma 1 proof compares semivalue weights while holding the least-core term fixed, reducing the signed-gap comparison to nonnegative summands.The argument uses stochastic dominance between context-size distributions.Source: §A.2 · p. 15
A.3 Consequences for classical semivalues
The appendix specializes the general gap result to classical semivalues and describes how context-size weights affect the signed gap. Under the stated hypotheses, shifting weight toward larger contexts decreases that gap for the considered families.
A.3 Consequences for classical semivalues
- The general gap theorem supplies explicit conditions for standard semivalues by substituting their context-size distributions.The weight-shifting lemma characterizes how the gap changes as those weights vary.Source: §A.3 · p. 15
- For Shapley, the grouped allocation is bounded below the least-core allocation by a term determined by sampled context-size marginal differences.A gap of at least δ is guaranteed when the stated sufficient condition on the probability of smaller contexts and the marginal-contribution difference holds.Source: §A.3 · p. 15
- For p-weighted Banzhaf, the actual signed gap is nonincreasing in p.When p ≤ 1/(m + 1), the result also gives a sufficient comparison with the Shapley δ-gap condition.Source: §A.3 · p. 16
- For Beta-Shapley, the signed gap is nondecreasing in a for fixed b and nonincreasing in b for fixed a.Thus increasing relative weight on larger contexts decreases the signed gap; Shapley is recovered at a = b = 1.Source: §A.3 · pp. 16–17
B.1 Epsilon evaluation for explanation methods
This appendix evaluates explanation stability through epsilon, the worst coalition deficit of an attribution vector. It compares post-hoc evaluations for several explanation methods with the optimized epsilon produced by the least-core solver.
B.1 Epsilon evaluation for explanation methods
- The evaluation uses a ResNet-50 on 100 ImageNet test images divided into 64 patch players each.Images are resized to 224 × 224 and split into an 8 × 8 grid.Source: §B.1 · p. 17
- Grad-CAM and Grad-CAM++ heat maps are converted to patch allocations and rescaled to satisfy efficiency before epsilon is evaluated.The maximum deficit is approximated over 4,096 sampled coalitions.Source: §B.1 · p. 17
- For Monte Carlo Shapley, epsilon is evaluated after estimating attributions from 64 random patch orderings per image.The procedure samples 4,096 output changes per image.Source: §B.1 · p. 17
- For least core, epsilon is the optimization objective over 4,096 sampled coalition constraints, rather than a post-hoc heat-map evaluation.The method uses 64 patch players and reports the optimized epsilon returned by the solver.Source: §B.1 · p. 17
- Lower epsilon indicates better satisfaction of the sampled core constraints and greater coalitional stability.Table 1 reports epsilon values on 100 ImageNet test images, with lower values better.Source: §B.1 · p. 17, Table 1 · p. 17
B.2.1 Main attribution comparison
In the controlled marker-bias experiment, the marker’s marginal effect reverses as non-marker context strengthens. Shapley and Banzhaf often assign the marker more attribution than the least-core, despite its harmful effect in strong contexts.
B.2.1 Main attribution comparison
- The controlled task adds a red marker to dog images and 30% of cat images, leaves birds unmarked, and evaluates cat-class logits with ResNet-50 and ViT-B/16.Images are divided into an 8 × 8 grid, and the marker-overlapping patches form D.Source: §B.2.1 · p. 17
- Mean low-/high-context marker contributions were 0.351/−0.373 for ResNet-50 and 0.917/−1.331 for ViT-B/16, reversing from help to suppression.The marker helps cat prediction in weak contexts but suppresses the cat logit when surrounding evidence is strong.Source: §B.2.1 · p. 18, Figure 4 · p. 18
- For the context test, 2,000 non-marker coalitions per image were sampled at probability 0.5, alongside the full non-marker context; selected images showed positive low-context and negative high-context effects.Attribution comparisons use 100 random permutations per image for Shapley and 100 marginal samples per patch for Banzhaf; the least-core uses 6,400 sampled constraints plus the empty and grand coalitions.Source: §B.2.1 · p. 18
- Shapley and Banzhaf assign more marker attribution than the least-core on average; the gap is pronounced for ViT-B/16 and Shapley on ResNet-50, but smaller for ResNet-50 Banzhaf.The comparison is per image, with the dashed line marking equal attribution.Source: Figure 5 · p. 18, §B.2.1 · p. 19
B.2.2 Effect of context weighting on the group semivalue gap
The experiment varies context-size weights for group semivalues while holding the model, masking, baseline, and least-core allocation fixed. Greater weight on larger contexts lowers the signed semivalue–least-core gap, sometimes past zero.
B.2.2 Effect of context weighting on the group semivalue gap
- The experiment treats the marker region D as one player and defines Gα so that a positive value means the group semivalue credits D more than the least-core does.The least-core reference is the original patch game’s allocation, rather than a sum of individual marker-patch semivalues.Source: §B.2.2 · p. 19
- The tested p-weighted Banzhaf settings were p = 0.2, 0.35, 0.5, 0.65, and 0.8, varying context-size distribution while model, masking, baseline, and least-core allocation remained fixed.The Beta-Shapley family includes (a, b) = (1, 1), which recovers group Shapley; an independently estimated group Shapley baseline was also used. Each group semivalue, including that baseline, used 3,200 marginal samples per image.Source: §B.2.2 · p. 19
- Weighted-Banzhaf gaps fell from 0.457 to −0.145 for ResNet-50 and 1.742 to −0.093 for ViT-B/16 as p rose from 0.2 to 0.8.For Beta-Shapley, shifting from (4, 1) to (1, 4) changed the gaps from 0.479 to −0.171 for ResNet-50 and from 1.331 to 0.069 for ViT-B/16.Source: §B.2.2 · p. 19, Figure 6 · p. 20
- The signed gap can cross zero at the largest context weights, so decreasing positive over-credit does not mean the absolute discrepancy keeps shrinking indefinitely.The stated mechanism is that larger contexts make it more likely to evaluate D where it is harmful.Source: §B.2.2 · p. 20
B.3 Contextual Distractors in Natural Images
On Salient ImageNet contextual distractors, Shapley attribution exceeds Core attribution across both architectures and region summaries, while random-SAM controls show the opposite tendency.
B.3 Contextual Distractors in Natural Images
- This appendix section reports additional evaluations of the Section 4 experiments on Salient ImageNet.Source: §B.3 · p. 20
- Mean AD gaps were 1.076 for ResNet-50 and 0.532 for ViT-B/16, with positive gaps in 82.5% and 80.6% of contextual-distractor candidates, respectively.The contrast remains under mask weighting; random-SAM controls instead have negative mean Shapley–Core gaps.Source: §B.3 · pp. 20–21
- Positive Shapley–Core gaps were enriched among regions with context-dependent sign reversal, rather than being a generic property of localized regions affecting prediction.The same pattern appears for both architectures and both region summaries.Source: §B.3 · pp. 20–21
B.4.1 Data, preprocessing, and cooperative game
The Salient ImageNet evaluation uses masked image patches as players and target-class logits as game values. Contextual distractors are selected through sign-reversal screening followed by independent localization and re-evaluation.
B.4.1 Data, preprocessing, and cooperative game
- Images and masks are resized and center-cropped to 224 × 224 pixels, split into 49 patches, and patches outside a coalition are replaced with the fixed ImageNet RGB mean before normalization.The game value is the target-class logit; the models are torchvision ResNet-50 with IMAGENET1K_V2 weights and ViT-B/16 with IMAGENET1K_V1 weights.Source: §B.4.1 · p. 21
B.4.2 Contextual-distractor selection
- Candidate features are screened as non-object regions by majority vote among five MTurk workers, then preliminary distractors are selected using binarized Salient ImageNet masks and patch overlap.The mask is thresholded at the image-specific 90th percentile; masks covering over 25% of the image are excluded, and a patch needs at least 20% overlap.Source: §B.4.2 · p. 21
- Candidates must have positive mean marginal contribution in weak contexts and negative contribution in strong contexts, with the corresponding sign occurring in a majority of sampled contexts in each group.Contexts are sampled by choosing context size uniformly and then a subset of that size uniformly.Source: §B.4.2 · p. 21
- Grounding DINO localization refined with SAM is checked against the Salient ImageNet mask, after which the region is redefined and the sign-reversal test repeated for each model’s own coalition values.The resulting contextual-distractor sets may differ across architectures.Source: §B.4.2 · p. 21
B.4.3 Shapley and sampled egalitarian least-core computation
The appendix describes sampled attribution procedures and prediction-active random-SAM controls, while limiting what can be inferred from comparing the resulting cohorts.
B.4.3 Shapley and sampled egalitarian least-core computation
- Shapley values use random player permutations, while sampled least-core constraints include independently sampled coalitions plus the empty and grand coalitions.The sampled egalitarian least-core fixes ε to ε⋆ within numerical tolerance, imposes no allocation nonnegativity, and is called Core in Section 4.2.Source: §B.4.3 · pp. 21–22
B.4.4 Prediction-active random-SAM controls
- Random-SAM controls use size-matched random boxes on images from the ImageNet pool, with prediction-active regions retained and sign-reversal regions excluded.Retention thresholds are 0.60 logit units for ResNet-50 and 0.90 for ViT-B/16; neither attribution method is used to select controls.Source: §B.4.4 · p. 22
B.4.5 Scope of the comparison
- The natural-distractor results characterize a selected population, and comparison with random-SAM controls is not a causal estimate of sign reversal alone.The natural cohort is selected using human annotations, independent localization, and a context-dependent sign-reversal criterion.Source: §B.4.5 · p. 22