Source-linked AI summary
Characterizing Structural Regularities of Labeled Data in Overparameterized Models
Ziheng Jiang, Chiyuan Zhang, Kunal Talwar, Michael C. Mozer
TL;DR
Deep networks must generalize across regular patterns while handling irregular examples, but instances can differ substantially in how readily they generalize. This paper measures per-instance consistency across training-set sizes and estimates scores for several image datasets. The scores distinguish regular from sparse or anomalous examples, while learning-speed proxies correlate better with them than distance-based proxies are reported to do.
Problem
Learning must capture common patterns while handling exceptions, and data manifolds may be multimodal rather than uniformly regular.
Source: §1 · p. 1Method
Results
C-scores characterize a continuum from dense, aligned modes to sparse modes, and learning-speed proxies generally correlate better than pairwise-distance proxies, which are metric-sensitive.
Source: §7 · p. 9Takeaways & Limitations
The authors demonstrate potential uses of C-scores for inspecting datasets and learning systems, identifying outliers, and analyzing learning dynamics across optimizers.
Source: §7 · p. 9C-scores depend on the architecture and learning algorithm; extensive comparison across more diverse architectures and emerging algorithms is left for future work.
Source: §7 · p. 9
Abstract
from arXiv · showhide
Humans are accustomed to environments that contain both regularities and exceptions. For example, at most gas stations, one pays prior to pumping, but the occasional rural station does not accept payment in advance. Likewise, deep neural networks can generalize across instances that share common patterns or structures, yet have the capacity to memorize rare or irregular forms. We analyze how individual instances are treated by a model via a consistency score. The score characterizes the expected accuracy for a held-out instance given training sets of varying size sampled from the data distribution. We obtain empirical estimates of this score for individual instances in multiple data sets, and we show that the score identifies out-of-distribution and mislabeled examples at one end of the continuum and strongly regular examples at the other end. We identify computationally inexpensive proxies to the consistency score using statistics collected during training. We show examples of potential applications to the analysis of deep-learning systems.
1. Introduction
Learning requires both generalizing regularities and memorizing exceptions, a balance also relevant to multimodal data and deep networks. The paper formalizes instance-level consistency across training-set sizes and uses it to study data regularities and learning dynamics.
1. Introduction
- Learning combines generalizing regular patterns with memorizing exceptions; because data manifolds are rarely unimodal, memorization is not always a failure.The boundary between regular and irregular examples can be fuzzy.Source: §1 · p. 1
- The consistency profile tracks expected correctness for a held-out instance as training-set size increases, capturing per-instance rather than average generalization.The definition focuses on multi-class classification and can extend to other problems by replacing 0-1 loss with a suitable loss.Source: §1 · pp. 1–2
- The C-score averages an instance’s consistency profile over training-set sizes, ordering examples along a continuum from regular to exceptional.High scores correspond to examples in well-supported regions; low scores can identify out-of-distribution or mislabeled instances.Source: §1 · p. 2
- The paper estimates C-scores across MNIST, CIFAR-10, CIFAR-100, and ImageNet, and finds learning-speed proxies correlate well while distance-based proxies are metric-sensitive.It presents the measure as a tool for analyzing data structure, learning dynamics, and potential applications such as outlier detection.Source: §1 · pp. 2–3
2. Related Work
Prior work studies data structure, prototypes, memorization, and example difficulty; this paper distinguishes its supervised, per-instance generalization focus and compares potential C-score proxies.
2. Related Work
- Unlike classical analyses of input or unsupervised representations, this work studies supervised learning and how data regularity interacts with overparameterized neural networks.In deep learning, the learner jointly learns a classifier and a representation capturing label semantics.Source: §2 · p. 3
- Carlini et al. (2018) study prototype measures such as confidence and learning speed, but their examples need not occupy the dense neighborhoods measured by high C-scores.Their ensemble agreement ignores labels and combines architectures with heterogeneous capacities.Source: §2 · p. 3
- The C-score extends related memorization and influence measures by examining a profile as training-set size increases, rather than one fixed-size evaluation.Feldman and Feldman & Zhang define memorization by the accuracy drop after removing a training example.Source: §2 · p. 3
- Learning speed correlates strongly with the C-score, whereas the paper’s distance-based proxies are sensitive to the underlying distance metric.Related work has examined learning speed in other contexts, including examples learned first and forgetting.Source: §2 · p. 3, Figure 2 · p. 3
3. The Consistency Profile and the C-score
The paper estimates per-instance consistency profiles from a fixed dataset and aggregates them into C-scores. In CIFAR, low-ranked examples can remain difficult even with large training subsets, while estimation uses sampled subsets and reuse to reduce computation.
3. The Consistency Profile and the C-score
- Because the underlying distribution is unknown, the empirical profile averages predictions from uniformly sampled training subsets that exclude the held-out example.For each training size, the estimator uses r independent subset samples.Source: §3 · p. 3
- Direct estimation is computationally prohibitive, so the method samples training sizes, interpolates between them, and reuses trained models across held-out examples.A naive CIFAR-10 estimate would require training more than 2 trillion models.Source: §3 · pp. 3–4
- Top-ranked CIFAR-100 examples are more likely than CIFAR-10 top examples to be classified correctly at low subset ratios.The authors sample subset ratios from 10% to 90% and use 2,000 subsets for empirical expectations.Source: §3 · p. 4
- Bottom-ranked CIFAR examples can remain below chance in correct-classification probability even at a 90% subset ratio, including mislabeled, ambiguous, or atypical cases.As the subset ratio grows, data regularities can systematically pull ambiguous instances toward the wrong prediction.Source: §3 · p. 4, Figure 3 · p. 4
- The C-score distills the consistency profile into a scalar by taking an expectation over training-set sizes.When sizes are sampled by subset ratio, the expectation is uniform over the sampled sizes.Source: §3 · p. 4
4. The Structural Regularities of Common Image Data Sets
Across common image datasets, C-score distributions show regular and rare, ambiguous examples, while the subset size that best approximates the integral score varies with dataset difficulty. On ImageNet, a point estimate supports class-level comparisons of score means and diversity.
4. The Structural Regularities of Common Image Data Sets
- Averaging C-scores over subset ratios reduces floor and ceiling effects; CIFAR-10’s integrated scores skew high but are more evenly spread than scores at individual ratios.Figure 4 compares subset-ratio histograms on CIFAR-10 with integrated-score histograms across three datasets.Source: §4 · p. 5, Figure 4 · p. 5
- Because full C-score estimation on ImageNet is computationally prohibitive, the authors use s = 70 and estimate scores from 2,000 ResNet-50 models trained on random 70% subsets.Training a standard ImageNet model costs one to two orders of magnitude more computing resources than training on CIFAR.Source: §4 · p. 5
- The best subset ratio is 10 for MNIST, 30 for CIFAR-10, and 40 or 50 for CIFAR-100, with the peak shifting higher as dataset challenge increases.The peak rank correlation identifies the subset ratio whose score ranking best matches the integral C-score ranking.Source: §4 · p. 5, Figure 5 · p. 5
- For ImageNet classes, mean C-score indicates relative difficulty and standard deviation indicates within-class diversity; the analysis plots their joint distribution and samples images across score percentiles.The selected images represent the top 99%, 35%, and 1% of within-class C-score rankings.Source: §4 · p. 5, Figure 5 · p. 5
- ImageNet classes commonly show a dense mode of regular examples and a tail of rare, ambiguous ones, with that tail smaller for upright and school buses than for car wheels.Projectile and yellow lady’s slipper illustrate extreme diverse and unified classes, respectively; most other classes lie in the histogram’s high-density region.Source: §4 · pp. 5–6
5. C-score Proxies
The paper seeks inexpensive proxies for the C-score, which is costly to estimate directly. Pairwise-distance proxies are sensitive to distance metrics, while learning-speed statistics correlate strongly with the C-score.
5. C-score Proxies
- The paper studies same-class and all-neighbor local-density scores, alongside LOF, as inexpensive alternatives to estimating C-score directly.A relative local-density score reflects whether an example is near same-label examples rather than far from them or near differently labeled examples.Source: §5 · p. 6, §5.1 · p. 6
5.1. Pairwise Distance Based Proxies
- Pairwise-distance rankings from raw pixels are not useful because the proxies lack sufficiently high correlation with C-score.LOF performs slightly better than the other tested input-space proxies.Source: §5.1 · p. 6, Table 1 · p. 6
- In learned hidden space, label-weighted distance ranks best, and appropriate label use improves ranking, but distance rankings remain sensitive to the underlying metric.The hidden representation retains some inter-example information even after training converges, yet the visualized rankings do not faithfully detect dense clusters of highly uniform, high-C-score examples.Source: §5.1 · p. 7
5.2. Learning Speed Based Proxies
- Learning-speed proxies are computationally trivial and use training-instance statistics, unlike the C-score, which concerns held-out instances after training.The authors motivate learning speed by the idea that consistent examples may be learned quickly because their gradient steps align.Source: §5.2 · p. 7
- ρ ≈0.9 at peak for pL, while the other learning-speed proxies also correlate strongly with C-score and forgetting events slightly underperform and peak later.The proxies use cumulative training statistics; pL is softmax confidence on the correct class.Source: §5.2 · pp. 7–8, Figure 7 · p. 7
6. Application
The C-score is used to examine data filtering, outlier detection, and optimizer learning dynamics. Its applications show both the value and risk of removing low-scoring examples, and expose differences between optimizer learning curves.
6. Application
- Removing the lowest C-score examples improves SVHN performance until too many are removed, when excluding correctly labeled rare examples causes performance to decline.On CIFAR-10, with fewer mislabeled examples, removing low-ranked examples offers no advantage over random removal.Source: §6 · p. 8
- 95.14% test accuracy for SGD versus 92.97% for Adam; SGD’s learning curves vary more across C-score groups, while Adam’s converge sooner.The authors suggest SGD’s stagewise schedule acts like a curriculum, focusing first on stronger regularities.Source: Figure 10 · p. 8, §6 · pp. 8–9
- For outlier detection, the evaluated forgetting-event statistic and hidden-space local outlier factor were not competitive.The comparison uses the forgetting statistic and LOF based on hidden-space distances.Source: §6 · p. 8
7. Discussion
The discussion presents the C-score as a per-instance measure of generalization and a ranking of examples by the training data needed for correct generalization. It reports dataset, proxy, and application findings while noting that scores depend on the learner.
7. Discussion
- The consistency profile tracks an individual example’s probability of correct generalization as training-set size grows, and the C-score ranks examples by estimated sample complexity.The authors report that this formulation captures intuitions about regularity in human and machine learning.Source: §7 · p. 9
- Across MNIST, CIFAR-10, CIFAR-100, and ImageNet, C-scores characterize a continuum from dense modes of aligned examples to sparse modes with only one or two instances.The reported dense modes contain aligned, centrally cropped examples with unified shape and color profiles.Source: §7 · p. 9
- Learning-speed proxies generally correlate better with the C-score than pairwise-distance proxies, which are sensitive to the underlying distance metric.Source: §7 · p. 9
- The C-score is presented as an analytical tool for inspecting datasets and learning systems, including identifying outliers and comparing optimizer learning dynamics.Source: §7 · p. 9
- C-scores depend on the neural-network architecture and learning algorithm; estimates across several common convolutional networks on CIFAR-10 nevertheless had Spearman’s ρ above 0.91.The authors leave extensive comparisons with more diverse architectures and emerging algorithms for future work.Source: §7 · p. 9
- The authors frame the work as progress toward understanding how consistency relationships among instances shape learning, and release precomputed benchmark-dataset scores for further research.Source: §7 · p. 9
Code and Pre-computed C-scores
The paper provides code, precomputed scores, and model checkpoints for CIFAR-10, CIFAR-100, and ImageNet, with dataset-specific information to map scores back to examples.
Code and Pre-computed C-scores
- The released code, precomputed C-scores, and associated model checkpoints cover CIFAR-10, CIFAR-100, and ImageNet.CIFAR score files include labels and scores in original dataset order; ImageNet scores are arbitrarily ordered and include a filename-based reconstruction script.Source: Code and Pre-computed C-scores · p. 9
A. Experiment Details
The appendix describes experimental settings and computational costs for C-score estimation, presents additional ranked-example visualizations, and specifies settings for pairwise-distance proxies.
A. Experiment Details
- C-score estimation uses holdout retraining on random training subsets; experiments used TensorFlow and mostly single P100 GPUs, with ImageNet jobs on eight P100 GPUs.The separate learning-speed experiments used ResNet-18 on CIFAR-10 for 200 epochs with batch size 32 and a stagewise learning-rate schedule.Source: A. Experiment Details · p. 11
B. Time and Space Complexity
- Holdout estimation has time complexity O(S(kT + E)) and score-computation space complexity O(kN); pairwise kernel-density scores require O(N^2) space and O(N^2d) time.Here S is the number of subset ratios, k the number of holdouts per ratio, T average network training time, and E the score-computation time.Source: B. Time and Space Complexity · pp. 11–12
C. More Visualizations of Images Ranked by C-score
- Additional figures visualize high-, middle-, and low-C-score examples from MNIST, CIFAR-10, CIFAR-100, and ImageNet.The CIFAR-100 visualization covers the first 60 of its 100 classes.Source: C. More Visualizations of Images… · p. 12
D. C-Score Proxies based on Pairwise Distances
- Pairwise-distance proxy experiments use an RBF kernel with bandwidth set to half the mean pairwise Euclidean distance and LOF with neighborhood size k = 3.Figure 16 shows LOF behavior across a range of neighborhood sizes.Source: D. C-Score Proxies based on Pairwise… · p. 12
D.1. Pairwise Distance Estimation with Gradient Representations
Gradient-based representations use examples’ loss gradients to rank their consistency, with similar gradients indicating consistency. They incorporate labels without ad hoc reweighting, but their high dimensionality makes naive density estimation computationally and memory intensive.
D.1. Pairwise Distance Estimation with Gradient Representations
- Similar gradients indicate that examples are consistent, so per-example gradients can serve as representations for density-based ranking.The gradient is evaluated at the network weights for each training input-label pair.Source: §D.1 · p. 12
- Loss gradients incorporate labels without ad hoc reweighting and can handle structured labels and losses such as image segmentation and machine translation.The loss already takes the label into account, so similar inputs with different labels produce dissimilar gradients.Source: §D.1 · pp. 12–13
- Figures 11 and 12 arrange MNIST and CIFAR-10 examples by class, with columns for high, intermediate, and low C-scores.Source: Figure 11 · p. 13, Figure 12 · p. 13
- Naive density estimation on gradient representations is prohibitive because modern networks can have millions to billions of parameters.The authors leave efficient algorithms for computing this score as future work.Source: §D.1 · p. 13
E. What Makes an Item Regular or Irregular?
Regularity reflects an example’s consistency with the population more than its intrinsic content, and is related to the model, loss, and learning algorithm. Equalizing C-score group sizes changes learning speeds, but the highest- and lowest-ranked groups remain separated.
E. What Makes an Item Regular or Irregular?
- High C-score groups contain more examples than low C-score groups, consistent with regularity arising from high probability mass.Source: E. What Makes an Item Regular or… · p. 13
- Above 30% accuracy near epoch 50 versus below 20% without subsampling, the lowest-ranked group learned faster after C-score groups were reduced to about 400 examples each.Many group learning curves also began to overlap after equalization.Source: E. What Makes an Item Regular or… · pp. 13, 16
- Figures 13–15 show class examples ranked by C-score for CIFAR-100 and ImageNet, with ImageNet panels also showing score histograms.Figure 13 presents the first 60 of CIFAR-100’s 100 classes; Figures 14–15 show five ImageNet classes.Source: Figure 13 · p. 14, Figure 14 · p. 15, Figure 15 · p. 15
- Higher-ranked groups learned more slowly after subsampling because the smaller dataset removed consistent examples that supported them.The authors contrast these lost supporters with the fewer competitors faced by lower-ranked groups.Source: E. What Makes an Item Regular or… · p. 16
- Even with overlapping learning curves, the highest- and lowest-ranked groups remained clearly separated; the authors suggest the highest-ranked examples may occupy fewer visual clusters.The lowest-ranked examples may be outliers in many different ways.Source: E. What Makes an Item Regular or… · p. 16
- Regularity arises from an example’s consistency with the population, not simply its intrinsic content, and depends on the model, loss function, and learning algorithm.A red lake and purple forest could be an outlier to a color-based model but regular to one that uses only grayscale information.Source: E. What Makes an Item Regular or… · p. 16
F. Sensitivity of C-scores to the Number of Models
The authors assess how many models are needed to estimate C-scores reliably. Correlations are good with fewer models for integral C-scores across subset ratios, while a fixed-ratio estimate requires more models for correlation above .95.
F. Sensitivity of C-scores to the Number of Models
- 512 models are needed for a fixed-subset-ratio C-score estimate to achieve > .95 correlation with the 1,000-model reference.The authors conclude that estimating integral or fixed-ratio C-scores requires on the order of 500–600 models.Source: F. Sensitivity of C-scores to the… · p. 16
- The two independent 1,000-model estimates correlate at 0.9996, and each correlates with the full 2,000-model estimate at 0.9999.Source: F. Sensitivity of C-scores to the… · p. 16
- With 64 models per subset ratio, estimated C-scores correlate well with the 1,000-model reference; Figure 20 plots this CIFAR-10 comparison with standard-deviation bars from 10 runs.The integral C-score spans nine subset ratios, so this entails roughly 64 × 9 models.Source: F. Sensitivity of C-scores to the… · p. 16, Figure 20 · p. 17