Soft-Label

Neuro-symbolic models are usually trained with supervision only on final labels, leaving the intermediate concepts unobserved. Since many concept assignments are consistent with a given label, training can predict labels correctly while recovering the wrong concepts, a failure known as a reasoning shortcut. Prototypical networks reduce shortcuts by anchoring each concept to a few labeled examples, but existing methods still couple perception and reasoning through a hand-crafted, task-specific differentiable loss that must be redesigned for every task. We introduce \textbf{Soft-PNet}, which removes this loss: it reframes concept grounding as a Metropolis walk over a precomputed cache of feasible symbolic solutions, guided by a prototype distribution built from a single labeled anchor per concept, and trains against one KL objective between the prototype-weighted cache and the network's concept predictions. The objective is identical across tasks and remains applicable when the solution space cannot be enumerated. On \texttt{MNIST-EvenOdd}, Visual Sudoku, and \texttt{Kand-Logic} under scarce supervision, Soft-PNet matches loss-engineered prototypical networks at the concept and label levels and recovers concepts that soft-grounding baselines miss, with no loss engineering and lower training time.

The LLM-as-a-judge (LLMaJ) framework offers a cost-effective and reproducible solution for automatic evaluation. However, current evaluation practices typically compare LLMaJ judgments against aggregated ground-truth labels, overlooking the valuable information contained in Human Label Variation (HLV). Inspired by an increasing line of work that proposes to leverage HLV, we systematically study LLMaJ performance on predicting both a single, aggregated ground truth hard-label and unaggregated soft-labels that represent Human Judgment Distributions (HJD). Our results across five diverse datasets reveal that while LLMs achieve near human-level performance at hard-label prediction on most tasks, they exhibit poor performance when predicting soft-labels. To address this limitation, we propose NAPHA (eNtropy-Aware Post-Hoc Alignment), a simple yet effective lightweight post-hoc alignment method that matches the LLM distribution to the HJD by first assigning an instance to a discrete entropy class and then routing it to specialized, trained alignment models. We find that NAPHA consistently improves soft-labels prediction across base LLM models and datasets, with particularly strong gains on high-entropy instances where capturing diverse human perspectives is most critical. We also show via oracle experiments that improving entropy class prediction can substantially enhance NAPHA's practical effectiveness.

Model selection for imbalanced binary classification often uses the Matthews correlation coefficient (MCC), but thresholding makes validation rankings threshold-dependent. SoftMCC is a post-training MCC validation framework on established probability-valued confusion counts, coupling an MCC-specific calibrated identity with a tie-aware, shared-pool selection protocol. Its core score is a covariance-normalized probability-label association, reduces exactly to MCC for hard predictions, and is Pearson-bounded. Under perfect population calibration it equals the Brier skill score with identical candidate ordering; outside that regime the gap does not identify calibration error. Across 18 settings with 12 duplicate-safe grouped repeats, SoftMCC attains the best stability mean rank (2.31) and highest mean tie-corrected Kendall's W (0.659), with a significant Friedman test (p=0.007); Nemenyi analysis separates it from AUPRC and MCC@0.5, while 14-source-family sensitivity retains only the latter. Selected-model utility shows no advantage. Three of six prespecified comparisons have negative mean test-MCC differences, only F1@best survives Holm correction (p=0.014), and the dataset-level test is not significant (p=0.117). Label permutation lowers mean W to 0.092; temperature scaling shifts SoftMCC rankings (mean Spearman 0.851) whereas rank-based and threshold-optimized metrics remain invariant. SoftMCC is a calibration-sensitive MCC-family selector with bounded stability and utility evidence.

This paper describes the BioSentinel team's participation in EXIST 2026 Task 2.2: Source Intention in Memes, part of the CLEF 2026 evaluation campaign. The task requires classifying the communicative intent behind memes as direct, judgemental, or no (non-sexist), under a Learning with Disagreement (Le-Wi-Di) paradigm that mandates both hard-label and soft-label (probability distribution) predictions. We present a text-centric approach built on xlm-roberta-base (270M parameters) trained with a composite loss function combining KL divergence on soft annotator distributions and weighted cross-entropy on hard labels. On the official test set, the system achieved an ICM-Soft-Norm of 0.3229 and ICM-Norm of 0.3778, with a hard F1-score of 0.4236, ranking 40th (out of 118 submissions) in the soft-soft evaluation and 49th (out of 187 submissions) in the hard-hard evaluation. We provide an analysis of the dataset characteristics, exploratory larger-architecture runs, and the role of annotator disagreement in shaping model design for subjective NLP tasks. Ablation results show that KL loss improves soft-label metrics, while CE loss improves hard-label accuracy. We also report a separate validation-set temperature analysis.

Infrared small target detection (IRSTD) commonly relies on pixel-level mask supervision. Such annotations, however, are costly and inherently uncertain because infrared targets have blurred boundaries and weak textures. We formulate box-supervised IRSTD as a problem distinct from generic box-to-mask segmentation and point-supervised IRSTD. Its central challenge is to construct stable pixel-level soft supervision from highly contaminated boxes. To this end, we propose Hotspot-Anchored Label Optimization (HALO). HALO localizes a radiometric anchor inside each box under local background-statistics constraints, then synthesizes a Physically Anchored Gaussian (PAG) soft label around the anchor. This turns noisy box supervision into continuous, pixel-level soft labels. The entire process is performed offline before training, remains decoupled from the detector backbone, and requires no online label updates. Experiments on public datasets show that HALO is competitive with representative box-supervised methods under standard tight boxes. Under looser or shifted box annotations that better approximate real scenarios, HALO is substantially more robust while remaining consistent across backbones. We further introduce a contamination-aware operating-regime analysis to characterize the effective boundary of this class of methods and reveal how intrinsic signal-to-clutter ratio relates to performance.

Temperature scaling is the dominant post-hoc calibration method in modern deep learning. Its theoretical justification rests on an assumption that is rarely stated explicitly: that ground-truth labels are one-hot and deterministic. In practice, labels are frequently soft, crowd-sourced, or genuinely distributional, reflecting real disagreement among human annotators rather than annotation noise. We study whether temperature scaling retains its calibration properties when this assumption is violated, and whether any resulting degradation depends on model scale. Using CIFAR-10H and ChaosNLI, two publicly available datasets with human-annotated soft label distributions, we evaluate three model scales per modality under both hard one-hot and soft distributional label targets. Across all nine configurations we find a positive soft-label calibration gap: temperature scaling calibrated on hard labels consistently underperforms an oracle calibrated directly on soft labels, with Brier Score gaps ranging from 0.002 to 0.134. The gap grows monotonically with model scale in the vision domain and on the SNLI-derived split of ChaosNLI, and is substantially larger in the language domain (mean gap 0.079) than in vision (mean gap 0.003). A scale-ordering reversal on the MNLI-derived split remains after matched-domain training; we treat it as inconclusive for the scale hypothesis and attribute it primarily to near-chance accuracy on that split. As a second post-hoc baseline, multiclass isotonic regression yields the same qualitative conclusion: positive soft-label gaps in all nine configurations, and larger gaps in language than in vision. These findings suggest that calibration protocols built on majority-vote labels systematically misstate model reliability wherever label ambiguity is structural, with direct consequences for deployment in safety-critical settings.

Data Shapley is the standard principled answer to which training points are worth what, and its k-nearest-neighbor (KNN) specialization is the version deployed in practice: the exact estimator shipped by toolkits such as pyDVL and OpenDataVal. Exact algorithms are known for unweighted KNN and for weighted KNN classification, but weighted KNN regression and soft-label prediction have resisted: the only exact method is an O(N^K) brute force, exponential in neighborhood size K. The obstruction: the weighted regression prediction is a ratio of two coalition-dependent sums, whose normalization denominator breaks the additive, threshold, and duplication structures the prior polynomial algorithms rely on. We close this gap. We give (i) the first pseudo-polynomial-time exact algorithm (polynomial in N and K at fixed lattice precision) for weighted KNN-regression Data Shapley, a counting dynamic program over the joint integer state (sum of w, sum of w*y), verified against exhaustive enumeration with zero mismatch on 12,716 adversarial instances; (ii) a certified FPTAS for continuous weights and targets, with a machine-checkable per-value error certificate never violated across 86,400 checks; (iii) a complexity landscape, including an unconditional Omega(D_w) output-size lower bound and access-model hardness results; and (iv) a weighted soft-label multi-class extension. We release an open-source, CPU-only library and the first exact weighted-regression Data Shapley ground truth. On downstream mislabel detection our exact values are statistically equivalent to Monte-Carlo Data Shapley (dataset-level TOST, n=8, p<10^-4), the pre-registered outcome; the value of exactness is instead determinism, a certified error bound, and an exact reference for auditing estimators: Monte-Carlo did not reproduce the exact top-10% ranking at any budget tested, up to 3,000 permutations (~1.28e6 utility evaluations).

Sign Language Retrieval (SLRet) enables efficient access to sign language content but remains fragile in fine-grained scenarios where visually similar signs must be distinguished. We show that this limitation does not stem from model capacity, but from ineffective hard negative supervision. Specifically, we formulate fine-grained retrieval failures as a negative distribution mismatch: semantically distinct yet visually confusable signs are rarely treated as hard negatives, while existing text-based mining strategies fail to capture such visual ambiguity. To address this issue, we propose Sign-Aware Hard Negative Mining (SAN), which constructs hard negatives based on visual confusability in the sign embedding space rather than linguistic similarity. Experiments on PHOENIX-2014T demonstrate that SAN substantially improves fine-grained retrieval performance while preserving coarse-grained accuracy, highlighting the importance of aligning negative supervision with visual ambiguity in sign language retrieval.

Subjective NLP tasks often exhibit systematic annotator disagreement, requiring models that represent uncertainty rather than collapse it. We introduce Ensemble Diversity Optimization (EDO), a prediction-space framework that jointly optimizes ensemble weights, effective cardinality, and calibration through a unified differentiable objective. EDO learns ensemble composition and size end-to-end via Gumbel-Softmax relaxation and incorporates a signed diversity regularizer, tuned on validation data, to steer optimization toward either preserving or suppressing disagreement. This regularization prevents ensemble collapse and enables controlled navigation of the utility-calibration trade-off. The framework integrates a soft F1 surrogate, class-weighted cross-entropy to address imbalance, and reliability-weighted diversity to regulate intra-ensemble variability. Experiments on four subjective text-classification benchmarks (ArMIS, ConvAbuse, HS-Brexit, MD-Agreement) show that EDO substantially improves probabilistic calibration, reducing cross-entropy (40-78% depending on baseline) and lowering Brier scores relative to Soft-CE, Soft-MD, Top-5 Voting, and WEL, while maintaining competitive F1 and better alignment with annotator distributions. These results demonstrate that jointly optimizing ensemble structure with a signed diversity regularizer provides an efficient, model-agnostic approach for modeling human subjectivity in supervised learning.

We present the AI Wizards submission to EXIST 2026 for multimodal sexism identification in memes. The task is composed of three, increasingly harder subtasks. We model them hierarchically as conditional soft-label prediction over empirical annotator distributions. Our system maps fixed Gemini Embedding 2 vision-language representations through a lightweight Gated MLP trained with KL divergence and homoscedastic uncertainty weighting. Our submissions ranked first on Task 2.3 and fourth on Tasks 2.1 and 2.2 on the official Soft-Soft leaderboards. The code is available at https://github.com/NLP-AI-Wizards/EXIST-2026

Speech emotion recognition (SER) often relies on hard consensus labels that collapse annotator disagreement. We study distribution-based supervision for 9-class SER on MSP-Podcast 2.0 using a WavLM-Base multitask model for categorical emotion and dimensional VAD. Hard-label training is compared with targets from primary and merged primary--secondary annotator vote distributions. Distributional objectives improve alignment with human vote distributions, reducing JSD/KLD relative to hard-label training. Analysis shows that hard supervision partly benefits from assigning ambiguous utterances to the residual Other class, whereas distributional supervision redistributes uncertainty across emotion categories. Entropy-stratified evaluation shows that high-ambiguity utterances remain challenging, but distribution-based supervision better captures perceptual uncertainty. These findings support moving beyond hard labels toward targets that reflect listener disagreement.

Emotions evolve through the dynamics of conversation, and understanding their transition structure is foundational to applications ranging from mental-health screening to dialogue systems. However, existing studies typically compress multi-rater judgments into a single hard label by majority voting, discarding the uncertainty signal needed to understand turn-to-turn transitions. In this article, we propose Bayesian Spectral Emotion Transition Discovery (BSETD), a two-stage framework that discovers emotion-transition structure from multi-rater soft labels. In the first stage, a hierarchical Dirichlet-Multinomial posterior is constructed through the outer product of soft labels, equipping each cell of the K x K transition matrix with a credible interval and Benjamini-Hochberg (BH) false discovery rate (FDR)-controlled significance. In the second stage, the symmetrized graph Laplacian is spectrally decomposed to separate a low-frequency (inertia) component from a high-frequency (contagion) component. On EmotionLines, BSETD simultaneously recovers the signatures of two distinct affective spaces: the Plutchik-adjacent transitions disgust to anger (log2 lift +0.94) and anger to disgust (+0.86) are over-represented, while the Russell-valence-reversed transitions joy to anger (-0.90) and anger to joy (-0.89) are under-represented. A five-source cross-corpus validation yields pairwise Pearson correlations in 0.91-0.98 within English, 0.79-0.85 against Chinese M3ED, and 0.979 between the human hard labels and the LLM virtual soft labels on the same utterance set, demonstrating that a pipeline preserving annotator uncertainty bridges the computational study of emotion dynamics with established psychological theory.

Large language models excel at general tasks but underperform smaller supervised models in specialized, high-stakes domains where training labels are costly. We address this regime with EvoPool, an evolutionary multi-agent framework inspired by Darwinian evolution. Three specialized agents iteratively propose executable annotator code, a small validation set provides a fitness signal, and a deterministic gate keeps only annotators that pass viability, diversity, and marginal-contribution checks across generations. Pool votes are mapped to soft training labels by EvoAgg, a text-aware aggregator combining semantic features with annotator-vote features. The authored pool runs at near-zero per-example cost and is 4500 to 31000x faster than LLM annotation on 100K examples. Across 7 of 8 LLM-weak specialized and complex tasks spanning biomedical relation extraction, legal-clause classification, complex reasoning, and dense multi-label biomedical classification, EvoPool beats the strongest LLM annotation baseline by an average +0.141 macro-F1, peaking at +0.301 on ChemProt and +0.265 on PubMed. Code is available at: https://github.com/tianyi0216/EvoPool

When annotators disagree on a label, the disagreement itself carries signal -- and the number of annotators needed to capture it depends on the evaluation metric. We fine-tune NLI models on label distributions subsampled from ChaosNLI, a dataset providing 100 independent annotator judgments per item, and identify metric-dependent saturation. In our 3-class NLI setting, entropy correlation -- whether the model identifies which items elicit disagreement -- requires N ~ 20-50 annotators to converge, while distributional match (KL divergence) saturates by N ~ 10 (87-95% of improvement across five model seeds). This finding rests on a prior observation: soft labels carry item-specific signal that label smoothing cannot replicate. Across five smoothing intensities, entropy correlation clusters at r ~ 0.45-0.49, while soft labels reach r = 0.643 (p < 0.001); per-item analysis traces this gap to smoothing's inability to distinguish ambiguous items from clear ones. The soft-label advantage replicates across two architectures (DeBERTa, RoBERTa), a non-NLI-pretrained baseline, and an exploratory cross-domain evaluation on content safety. These results suggest that annotation budgets should be informed by the target evaluation metric rather than set uniformly.

Background and objective. Conventional Deep Learning (DL) approaches for Knee Osteoarthritis (KOA) grading rely on one-hot labels, which fail to capture both the ordinal uncertainty of Kellgren--Lawrence (KL) and Calcium Pyrophosphate Deposition Disease (CPPD) severity scores and the asymmetric relationship between the two scales observed in clinical practice. Methods. We retrospectively collected 2172 knee X-ray images, including 968 radiographs jointly annotated for KL and CPPD severity. An ordinal DL framework based on soft-labelling was developed for both tasks, replacing one-hot targets with unimodal probability distributions centred on the annotated grade. Four formulations were investigated: binomial, beta, triangular, and exponential. Results. All soft-labelling strategies consistently outperformed the nominal baseline. For CPPD grading, the triangular formulation achieved the highest Quadratic Weighted Kappa (QWK) and the lowest Mean Absolute Error (MAE) (QWK = 0.796; MAE = 0.438), while the beta formulation yielded the most balanced class-wise performance considering Average MAE (AMAE) and Maximum MAE (MMAE) across classes (AMAE = 0.458; MMAE = 0.573). For KL grading, the beta-based approach provided the best overall performance, achieving the highest QWK together with the lowest MAE and class-wise errors (QWK = 0.777; MAE = 0.529; AMAE = 0.523; MMAE = 0.775). Statistical analysis demonstrated significant improvements over conventional one-hot supervision (p < 0.001).

Knowledge distillation (KD) transfers knowledge from a large teacher model to a smaller student. In language modeling, the student is trained either on tokens sampled from the teacher (hard labels) or the teacher's full next-token distribution (soft labels). Despite soft labels appear strictly richer, we find that mixing hard and soft labels consistently yields better results. Crucially, we show that this gain cannot be explained by closer teacher matching during training. Instead, it comes from reduced exposure bias, the mismatch between training and inference distributions. To explain this phenomenon, we introduce the Bridge-Garden Decomposition theory, which categorizes generation steps into two types: Bridges, where the next token must be exact, and Gardens, where it can be flexible. We show that hard-only KD excels in Bridges by avoiding risky deviations, while soft-only KD preserves diversity in Gardens. A hybrid strategy handles both cases and, as a result, reduces exposure bias across the sequence. Guided by this theory, we develop a family of Bridge-Garden hybrid supervision methods that adaptively balance hard and soft labels. Across a primary suite of seven teacher-student pairs (including Qwen, Llama, Gemma, and DeepSeek) and benchmarks in reasoning and coding, our approach outperforms divergence-based and on-policy KD baselines while reducing training cost by 9.7x, enabling efficient model compression. Code is available at https://github.com/ghwang-s/bridge_garden_hybrid_kd_release.

Annotator disagreement in emotion classification reflects ambiguity intrinsic to emotion concepts and is essential for predictor-quality assessment in subjective NLP. Yet no prior work integrates soft-label learning with Bayesian deep learning to evaluate uncertainty along axes including annotator-distribution fidelity. We train a linear head on a frozen RoBERTa via cyclical stochastic gradient Markov chain Monte Carlo (cSG-MCMC), targeting the empirical annotator distribution with a soft-label objective under a five-axis evaluation. On the 28-emotion GoEmotions benchmark, the proposed method outperforms Monte Carlo Dropout and Deep Ensemble simultaneously on three axes -- Jensen-Shannon divergence (JSD) to the annotator distribution, Spearman correlation between per-emotion aleatoric uncertainty and disagreement, and selective-prediction Area Under the Risk-Coverage Curve (AURC) and Area Under the ROC Curve (AUROC) -- showing independent axes are jointly attainable from one posterior. Post-hoc temperature scaling exhibits a bidirectional effect, establishing hard-label calibration and annotator-JSD as independent dimensions and motivating joint reporting as an honest protocol.

Metagenomic taxonomic annotation aims to identify the microbial origins of DNA fragments in environmental samples. Traditional methods that rely on sequence similarity are often constrained by the high microbial diversity and the incompleteness of reference databases, which has motivated the development of learning approaches such as Taxometer that perform post hoc correction to learn more informative metagenomic sequence representations. However, these methods typically rely on labels derived from similarity search tools during training, which inevitably introduces noise that can impair representation learning and degrade classification performance. To address this issue, we propose TaxDistill, a knowledge distillation framework for metagenomic classification. We introduce GenomeOcean, a 500M parameter genomic foundation model, as the teacher network to extract deep semantic features and generate soft labels based on confidence. By distilling this soft label information into a lightweight student network, TaxDistill effectively reduces the label noise introduced by initial retrieval tools. Comprehensive experiments on seven diverse CAMI2 datasets demonstrate that TaxDistill outperforms existing baselines in most scenarios. For instance, on the Gastrointestinal dataset, it improves the F1 score of MMseqs2 from 0.763 to 0.941, outperforming the Taxometer baseline. Overall, TaxDistill provides a reliable method for label correction in complex metagenomic analysis.

When annotators disagree, that disagreement can reflect epistemic uncertainty rather than simple label noise. We study hard-label delivery as an alternative to the usual choices of collapsing votes to a single label or training directly on the empirical soft-label distribution. We focus on two primary hard-label methods: multipass, which cycles through observed votes while keeping the dataset size fixed, and stochastic label sampling (SLS), which samples one label per example at the start of each epoch. On CIFAR-10H, we find that when only a small number of annotations per example is available, hard-label delivery improves over soft-label training, with larger improvements where the sparse empirical target is farther from the full annotator distribution. When full annotator distributions are available, both hard-label methods match soft-label training. We use deterministic control as an ablation of multipass and shuffled SLS as a control that breaks the example-to-distribution match. We also show that SLS and soft-label cross-entropy optimize the same expected objective. Hard-label delivery also converges to flatter basins, with supporting descriptive evidence from OOD detection on SVHN and CIFAR-100. Overall, these results suggest that multipass is a strong practical default when raw vote counts are available, while SLS offers a lightweight alternative that remains competitive when only a few votes per example are available and matches soft-label training when full annotator distributions are available.

Central to human-aligned AI is understanding the benefits of human-elicited labels over synthetic alternatives. While human soft-labels improve calibration by capturing uncertainty, prior studies conflate these benefits with the implicit correction of mislabeled data (mode shifts), obscuring true effects of soft-labels. We present a controlled audit of soft-label learning across MNIST and a synthetic variant, re-annotating subsets to extract human uncertainty. By decoupling soft-label supervision from underlying label mode shifts, we show that while human soft-labels do provide accuracy gains, their larger value lies in acting as a regularizer that improves model calibration on difficult samples and promotes stable convergence across training runs. Dataset cartography reveals models trained on human soft-labels mirror human uncertainty, whereas those trained on synthetic labels fail to align with humans. Broadly, this work provides a diagnostic testbed for human-AI uncertainty alignment.

Accurately identifying student misconceptions is crucial for personalized education but faces three challenges: (1) data scarcity with long-tail distribution, where authentic student reasoning is difficult to synthesize; (2) fuzzy boundaries between error categories with high annotation noise; (3) deployment parado-large models overlook unconventional approaches due to pretraining bias and cannot be deployed on edge, while small models overfit to noise. Unlike traditional methods that increase diversity through large-scale data synthesis, we propose a two-stage knowledge distillation framework that mines high-value samples from existing data. The first stage performs standard distillation to transfer task capabilities. The second stage introduces a dual-layer marginal selection mechanism based on cognitive uncertainty, identifying four types of critical samples based on teacher model uncertainty and confidence differences. For different data subsets, we design difficulty-adaptive mechanism to balance hard/soft label contributions, enabling student models to inherit inter-class relationships from teacher soft labels while distinguishing ambiguous error types. Experiments show that with augmented training on only 10.30% of filtered samples, we achieve MAP@3 of 0.9585 (+17.8%) on the MAP-Charting dataset, and using only a 4B parameter model, we attain 84.38% accuracy on cross-topic tests of middle school algebra misconception benchmarks, significantly outperforming sota LLM (67.73%) and standard fine-tuned 72B models (81.25%). Our code is available at https://github.com/RoschildRui/acl2026_map.

Despite the perceived success of large-scale dataset distillation (DD) methods, recent evidence finds that simple random image baselines perform on-par with state-of-theart DD methods like SRe2L due to the use of soft labels during downstream model training. This is in contrast with the findings in coreset literature, where high-quality coresets consistently outperform random subsets in the hardlabel (HL) setting. To understand this discrepancy, we perform a detailed scalability analysis to examine the role of data quality under different label regimes, ranging from abundant soft labels (termed as SL+KD regime) to fixed soft labels (SL) and hard labels (HL). Our analysis reveals that high-quality coresets fail to convincingly outperform the random baseline in both SL and SL+KD regimes. In the SL+KD setting, performance further approaches nearoptimal levels relative to the full dataset, regardless of subset size or quality, for a given compute budget. This performance saturation calls into question the widespread practice of using soft labels for model evaluation, where unlike the HL setting, subset quality has negligible influence. A subsequent systematic evaluation of five large-scale and four small-scale DD methods in the HL setting reveals that only RDED reliably outperforms random baselines on ImageNet-1K, but can still lag behind strong coreset methods due to its over-reliance on easy sample patches. Based on this, we introduce CAD-Prune, a compute-aware pruning metric that efficiently identifies samples of optimal difficulty for a given compute budget, and use it to develop CA2D, a compute-aligned DD method, outperforming current DD methods on ImageNet-1K at various IPC settings. Together, our findings uncover many insights into current DD research and establish useful tools to advance dataefficient learning for both coresets and DD.