Diversity and Difficulty Aware Sampling

• Diversity-and-difficulty-aware sampling is a method that combines coverage (diversity) and hardness (difficulty) criteria to improve selection in various domains.
• It quantifies diversity through metrics like embedding spread and cluster-periphery distance, and measures difficulty using metrics such as F1-derived error rates and uncertainty scores.
• Empirical results show that jointly optimizing for diversity and difficulty enhances model performance, sample efficiency, and training dynamics across tasks.

Searching arXiv for papers on diversity-and-difficulty-aware sampling and closely related formulations. Diversity-and-difficulty-aware sampling denotes a class of sampling, pruning, or decoding procedures in which selection is guided simultaneously by a coverage criterion and a hardness criterion. In the cited literature, this coupling appears in data-centric augmentation for image classification, graph-based coreset pruning, rate-distortion and Determinantal Point Process selection, instruction-tuning subset construction, selective language-model decoding, active learning, dynamic Direct Preference Optimization for mathematical reasoning, and multi-armed-bandit selection of generated tabular batches (Cavusoglu et al., 2021, Maharana et al., 2023, Chen et al., 2023, Zhang et al., 14 Mar 2025, Troshin et al., 20 Sep 2025, Arya et al., 21 May 2026, Rao et al., 22 May 2025, Tang et al., 26 Dec 2025).

1. Representative formalizations

The literature does not use a single universal definition. Instead, it instantiates “diversity” as embedding-space spread, coreset coverage, semantic separation, cluster-periphery distance, or output distinctness, while “difficulty” appears as complement of per-class $F1$, training-dynamics importance, uncertainty-mitigated prediction loss, decoding risk, posterior uncertainty, or self-aware failure propensity.

Work	Diversity term	Difficulty term
Iterative sampling (Cavusoglu et al., 2021)	$\mathrm{Div}(D_c)$ from minimal inter-sample distances in a ResNet50 embedding	$\mathrm{Diff}_c = 1 - F1_c$
D$^2$ Pruning (Maharana et al., 2023)	$k$-NN graph structure and reverse message-passing downweighting	node difficulty $x_i = f_\theta(s_i)$ from training dynamics
RD-DPP (Chen et al., 2023)	$R(Z,\epsilon)$ and class-conditional $sdiv(Z)$	uncertainty by cross-entropy or margin after the phase switch
D$^3$ (Zhang et al., 14 Mar 2025)	$d_1(z,Q)=\min_{z'\in Q} d(z,z')$	uncertainty-based prediction difficulty $\mathrm{Div}(D_c)$0
Selective sampling (Troshin et al., 20 Sep 2025)	diversity evaluated by avg. distinct $\mathrm{Div}(D_c)$1-gram over the subset of correct samples	sampling risk $\mathrm{Div}(D_c)$2
LCD / DSAL (Arya et al., 21 May 2026)	cluster-center distance $\mathrm{Div}(D_c)$3	least confidence $\mathrm{Div}(D_c)$4
SAI-DPO (Rao et al., 22 May 2025)	clustering of knowledge-point embeddings $\mathrm{Div}(D_c)$5 into categories $\mathrm{Div}(D_c)$6	ordered by $\mathrm{Div}(D_c)$7, number of chain-of-thought steps, and response length
DATE / MDS (Tang et al., 26 Dec 2025)	$\mathrm{Div}(D_c)$8 from weighted rule overlap	quality $\mathrm{Div}(D_c)$9, with a stated difficulty-aware extension $\mathrm{Diff}_c = 1 - F1_c$0

2. Diversity as coverage, spread, and semantic separation

A basic formulation appears in iterative augmentation for classification. Let $\mathrm{Diff}_c = 1 - F1_c$1 be the current set of training samples of class $\mathrm{Diff}_c = 1 - F1_c$2, let $\mathrm{Diff}_c = 1 - F1_c$3 denote the embedding of sample $\mathrm{Diff}_c = 1 - F1_c$4 extracted from the global-average-pooling layer of a ResNet50, and define pairwise Euclidean distance by

$\mathrm{Diff}_c = 1 - F1_c$5

The intra-class diversity metric is then

$\mathrm{Diff}_c = 1 - F1_c$6

so a larger $\mathrm{Diff}_c = 1 - F1_c$7 means that a class’s samples are more spread out in embedding space. The same work also defines augmentation-induced spread through $\mathrm{Diff}_c = 1 - F1_c$8, averaging the variance of augmented embeddings around their mean, thereby treating diversity as a property of both the retained set and the augmentation pipeline (Cavusoglu et al., 2021).

Coreset-oriented formulations typically encode diversity as distance to what has already been selected. In D$\mathrm{Diff}_c = 1 - F1_c$9, for a full instruction-response pool $^2$0 and a coreset $^2$1, the diversity function is

$^2$2

where $^2$3 is cosine distance between sentence embeddings $^2$4 extracted by the current LLM $^2$5. A large $^2$6 indicates that a sample brings new coverage, and the paper explicitly interprets this as approximating a $^2$7-center covering of the full pool (Zhang et al., 14 Mar 2025).

Graph-based pruning encodes diversity structurally rather than purely metrically. D$^2$8 Pruning represents the dataset as an undirected weighted graph whose edges connect $^2$9 nearest neighbors under embedding distance $k$0. Once a node $k$1 is selected, reverse message passing updates neighboring scores as

$k$2

so nearby samples are downweighted more strongly than distant ones. The stated role of this step is to enforce spatial diversity during coreset construction (Maharana et al., 2023).

A more explicitly task-oriented semantic notion appears in RD-DPP. For data matrix $k$3, the rate-distortion proxy is

$k$4

with $k$5. For a $k$6-class task with class index sets $k$7, the paper defines

$k$8

thereby measuring the net gain of overall diversity minus average within-class compressibility. In that framework, diversity is not merely pairwise dispersion but explicitly class-conditional semantic separation (Chen et al., 2023).

Other domains adopt lighter-weight proxies. In the active-learning framework containing LCD, feature vectors are clustered and diversity is ranked by distance from the cluster center,

$k$9

where large $x_i = f_\theta(s_i)$0 indicates that $x_i = f_\theta(s_i)$1 lies on the periphery of its cluster. In SAI-DPO, problem diversity is mediated by “knowledge-point” descriptors tagged by a strong annotator LLM, embedded by a pre-trained Sentence-Transformer, and clustered by $x_i = f_\theta(s_i)$2-means into categories $x_i = f_\theta(s_i)$3; sampling across clusters is treated as sampling across knowledge coverage. In DATE, diversity at the final sampling stage is defined by weighted overlap between a generated batch’s Distribution-Guiding Rule and the original prompt-example rules, using Jaccard overlap over rule predicate sets (Arya et al., 21 May 2026, Rao et al., 22 May 2025, Tang et al., 26 Dec 2025).

3. Difficulty as error propensity, uncertainty, and model-state dependence

The simplest hardness definition in the cited literature is class-level and validation-based. In iterative augmentation, per-class precision and recall on the held-out validation set yield

$x_i = f_\theta(s_i)$4

Classes with larger $x_i = f_\theta(s_i)$5 are deemed harder, and this score is later used to rank classes and to identify $x_i = f_\theta(s_i)$6 for additional edge-case mining (Cavusoglu et al., 2021).

Instruction-tuning selection requires a sample-level notion that is less sensitive to open-ended generation variability. D$x_i = f_\theta(s_i)$7 therefore introduces Uncertainty-based Prediction Difficulty (UPD). For sample $x_i = f_\theta(s_i)$8 with tokenized response $x_i = f_\theta(s_i)$9, the base per-token cross-entropy is $R(Z,\epsilon)$0, the predictive entropy is

$R(Z,\epsilon)$1

and the token score is

$R(Z,\epsilon)$2

The sample-level difficulty is then $R(Z,\epsilon)$3. The stated intent is to downweight tokens with high loss but also high entropy, so that “truly hard” tokens for instruction alignment remain emphasized (Zhang et al., 14 Mar 2025).

Several works define difficulty directly in terms of uncertainty. RD-DPP switches, after diversity saturation, to uncertainty-based mode using either

$R(Z,\epsilon)$4

or the margin between the top two class probabilities, treating high uncertainty or small margin as classification difficulty. The LCD framework centers on least confidence,

$R(Z,\epsilon)$5

while also noting that the same framework can accommodate margin or entropy in place of $R(Z,\epsilon)$6 (Chen et al., 2023, Arya et al., 21 May 2026).

For generative decoding, difficulty can be localized to a token position rather than to an entire example. Selective sampling defines a sampling-risk metric at decoding prefix $R(Z,\epsilon)$7:

$R(Z,\epsilon)$8

If this quantity is large, high-temperature sampling at that position is likely to reduce final accuracy relative to greedy continuation. Difficulty is therefore the risk of sampling itself, estimated by a lightweight classifier on the base LM’s hidden states (Troshin et al., 20 Sep 2025).

Dynamic training introduces a further shift: difficulty becomes model-stage dependent. SAI-DPO defines a three-stage ordering on problems using $R(Z,\epsilon)$9 with $sdiv(Z)$0 rollouts, then number of generated chain-of-thought steps, then response length. Lower $sdiv(Z)$1 implies harder; ties are broken first by more steps and then by longer responses. The algorithm then builds an “error set” consisting of the 50% hardest problems among those the model neither always solves nor always fails. DATE makes the same point more indirectly: its final Multi-Armed Bandit selection is originally formulated for diversity and quality, but the paper states that the same framework can be repurposed to target “difficulty” instead of “quality” by replacing $sdiv(Z)$2 with a per-arm uncertainty score $sdiv(Z)$3 (Rao et al., 22 May 2025, Tang et al., 26 Dec 2025).

4. Sampling algorithms and optimization patterns

One recurring pattern is iterative replacement. In the image-classification framework of iterative diversity-aware sampling, a current training set $sdiv(Z)$4 and an augmentation pool $sdiv(Z)$5 are maintained per class. At each iteration, the algorithm finds all samples whose embedding distance to some other in-class sample falls below a threshold $sdiv(Z)$6, removes these “duplicate” samples, and refills the class by randomly sampling from the remaining pool until the target class size $sdiv(Z)$7 is restored. After this remove-and-replace loop, hard classes are further expanded by visually inspecting misclassified validation examples and selecting pool augmentations nearest to the misclassified embeddings,

$sdiv(Z)$8

up to a budget $sdiv(Z)$9 (Cavusoglu et al., 2021).

A second pattern is weighted coreset construction. D$^3$0 formulates subset selection as

$^3$1

which is presented as a weighted $^3$2-center problem and therefore NP-hard. The practical solver is greedy: iteratively add the sample that maximizes the current weighted distance to the already selected set. D$^3$3 Pruning also uses a greedy loop, but after a forward message pass that updates node scores by aggregating weighted difficulty from neighbors. Its reverse message pass then depresses nearby scores after each selection, thereby coupling local difficulty context and global coverage in one priority-queue procedure (Zhang et al., 14 Mar 2025, Maharana et al., 2023).

A third pattern is staged switching. RD-DPP begins in a “diversify” phase, computes semantic-diversity gains $^3$4 for candidates, forms a DPP kernel, and performs greedy DPP-MAP selection. Once the marginal gain in semantic diversity falls below the threshold $^3$5, the algorithm switches to an “uncertainty” phase and selects the top-$^3$6 most uncertain samples. The paper describes this as a response to the observed phase transition in DPP diversity gain (Chen et al., 2023).

In LLM inference and training, the same principle appears in online form. Selective sampling computes a risk score $^3$7 at each decoding step from the base LM’s hidden states; if $^3$8, with $^3$9 typically set to $d_1(z,Q)=\min_{z'\in Q} d(z,z')$0, decoding is greedy, otherwise it samples from a high-temperature truncated distribution such as min-$d_1(z,Q)=\min_{z'\in Q} d(z,z')$1 with $d_1(z,Q)=\min_{z'\in Q} d(z,z')$2. LCD applies a two-stage active-learning pipeline: rank the unlabeled pool by uncertainty, retain the top-$d_1(z,Q)=\min_{z'\in Q} d(z,z')$3 most uncertain points, compute feature-space cluster-center distance on that candidate pool, and then select the top-$d_1(z,Q)=\min_{z'\in Q} d(z,z')$4 by diversity. SAI-DPO alternates between a 1% stratified probe phase, which updates error-weighted cluster probabilities using current model performance, and an Iterative Preference Optimization phase, which filters out trivially solved and trivially failed items, constructs $d_1(z,Q)=\min_{z'\in Q} d(z,z')$5 triplets, removes the easiest 30%, retains the top 70% by self-aware difficulty, and updates the policy by the DPO objective. DATE’s MAB-based Data Sampling uses a “Successive Accepts and Rejects” style procedure: each candidate generated batch is an arm, pulls re-estimate empirical reward $d_1(z,Q)=\min_{z'\in Q} d(z,z')$6, and low-reward arms are progressively rejected under a total pull budget $d_1(z,Q)=\min_{z'\in Q} d(z,z')$7 (Troshin et al., 20 Sep 2025, Arya et al., 21 May 2026, Rao et al., 22 May 2025, Tang et al., 26 Dec 2025).

5. Empirical behavior across domains

The reported gains span data augmentation, coreset pruning, active learning, instruction tuning, decoding, reasoning, and generated-data filtering.

Method	Setting	Reported result
Iterative sampling (Cavusoglu et al., 2021)	raw $d_1(z,Q)=\min_{z'\in Q} d(z,z')$8 uneven on the Roman-numeral competition	Val Acc. $d_1(z,Q)=\min_{z'\in Q} d(z,z')$9; biggest jump at “iter” $\mathrm{Div}(D_c)$00; cumulative gain $\mathrm{Div}(D_c)$01
D$\mathrm{Div}(D_c)$02 Pruning (Maharana et al., 2023)	ImageNet-1K at $\mathrm{Div}(D_c)$03 pruning	$\mathrm{Div}(D_c)$04 pp over Coverage-centric Coreset Selection
RD-DPP (Chen et al., 2023)	CIFAR-10 across EfficientNet-B0, ResNet-18, ResNeXt	consistently outperforms all baselines by $\mathrm{Div}(D_c)$05–$\mathrm{Div}(D_c)$06, especially at low budgets
D$\mathrm{Div}(D_c)$07 (Zhang et al., 14 Mar 2025)	Alpaca (52K) with only $\mathrm{Div}(D_c)$08 selected	average “winning score” $\mathrm{Div}(D_c)$09 vs. full-data fine-tuning; AlpacaEval $\mathrm{Div}(D_c)$10 vs. $\mathrm{Div}(D_c)$11 for full
Selective sampling (Troshin et al., 20 Sep 2025)	Quality–Diversity AUC on GSM8K / Symbolic / Minerva	$\mathrm{Div}(D_c)$12
LCD (Arya et al., 21 May 2026)	CIFAR-10 / ResNet-18; PASCAL VOC / VGG-16	$\mathrm{Div}(D_c)$13 vs Core-set $\mathrm{Div}(D_c)$14; $\mathrm{Div}(D_c)$15 vs CDAL-RL $\mathrm{Div}(D_c)$16
SAI-DPO (Rao et al., 22 May 2025)	Eight mathematical reasoning benchmarks	average boost $\mathrm{Div}(D_c)$17 pts; AIME24 $\mathrm{Div}(D_c)$18; AMC23 $\mathrm{Div}(D_c)$19 pts–$\mathrm{Div}(D_c)$20 pts
DATE / MDS (Tang et al., 26 Dec 2025)	Eight classification tasks; two regression tasks	average error $\mathrm{Div}(D_c)$21; average MSE $\mathrm{Div}(D_c)$22

The empirical patterns are consistent with the algorithms’ design choices. In the Roman-numeral competition, the largest single jump came from the iterative remove-and-replace stage, which the paper states verifies that removing embedding-duplicates and back-filling with diverse augmentations substantially improves generalization; a further “uneven” class-size skew toward harder classes provided an additional modest boost (Cavusoglu et al., 2021). In selective decoding for mathematical reasoning, the quality–diversity frontier improves most clearly in high-temperature regimes: at $\mathrm{Div}(D_c)$23, baselines lose accuracy sharply, whereas selective sampling maintains $\mathrm{Div}(D_c)$24–$\mathrm{Div}(D_c)$25 pp higher accuracy for the same diversity, and yields $\mathrm{Div}(D_c)$26–$\mathrm{Div}(D_c)$27 lower perplexity than min-$\mathrm{Div}(D_c)$28 across temperatures (Troshin et al., 20 Sep 2025).

For sample-efficient instruction tuning, D$\mathrm{Div}(D_c)$29 reports that performance peaks earlier, for example at $\mathrm{Div}(D_c)$30–$\mathrm{Div}(D_c)$31 of the full pool, and then gracefully degrades toward full-data performance as the budget increases. In graph-based pruning, D$\mathrm{Div}(D_c)$32 improves over either geometry-only or difficulty-only baselines, including at high pruning rates and in self-supervised and multimodal settings. In active learning, LCD is reported to outperform Core-set, VAAL, Influence AL, and the other three hybrid samplers across all evaluated model–dataset pairs, while DSAL performs best at a 50:50 split of hard-diverse and easy-diverse samples (Zhang et al., 14 Mar 2025, Maharana et al., 2023, Arya et al., 21 May 2026).

Dynamic model-aware selection yields a different empirical signature: faster convergence and changing target regions over time. SAI-DPO reports that it reaches peak performance in fewer total samples than random-IDPO and that the evolving error profile shows more hard problems being resolved per iteration. DATE’s MAB stage is motivated by an analogous observation in heterogeneous tabular generation: the paper argues that validation-best greedy selection can miss arms that jointly improve performance, and its MDS procedure improves both classification error and regression MSE over greedy baselines (Rao et al., 22 May 2025, Tang et al., 26 Dec 2025).

6. Misconceptions, limitations, and open directions

A recurring misconception is that diversity and difficulty can be optimized independently without loss. D$\mathrm{Div}(D_c)$33 Pruning states the counterpoint explicitly: optimizing for data diversity leads to a coreset biased toward easier samples, whereas selection by difficulty ranking omits easy samples necessary for training deep learning models. RD-DPP reaches a related conclusion from a different route: DPP is beneficial only at the beginning of sample accumulation, and pure RD-DPP suffers after the transition point, which is why the method switches to uncertainty mode once semantic-diversity gains saturate (Maharana et al., 2023, Chen et al., 2023).

Another misconception is that difficulty is a fixed external property of a sample. The cited works repeatedly define it relative to a model, a training phase, or a decoding state. Selective sampling uses prefix-specific risk under high-temperature sampling; SAI-DPO reports that static “external” difficulty attains an average of $\mathrm{Div}(D_c)$34, worse than the baseline $\mathrm{Div}(D_c)$35, while the full self-aware metric reaches $\mathrm{Div}(D_c)$36; DATE states that quality can be replaced by difficulty in the reward, with the trade-off controlled by $\mathrm{Div}(D_c)$37 in a combined objective. This suggests that difficulty is often model- and stage-dependent rather than invariant (Troshin et al., 20 Sep 2025, Rao et al., 22 May 2025, Tang et al., 26 Dec 2025).

The methods also inherit concrete operational constraints. Iterative augmentation relies strongly on the fidelity of augmented samples and the diversity of augmentation methods. D$\mathrm{Div}(D_c)$38 Pruning requires a pretrained or fully trained model to derive initial difficulty scores and embeddings and therefore cannot be applied “cold,” though self-supervised or unsupervised proxies may help. Selective sampling uses a model-specific classifier head, is not immediately portable across different LLM weights, is focused on verifiable math reasoning, and evaluates diversity by $\mathrm{Div}(D_c)$39-gram distinctness rather than richer metrics. LCD notes that clustering can become expensive for very large pools and currently uses a single uncertainty metric. DATE further proves that greedy-choice fails in heterogeneous generated-data selection, so the final sampling step is not reducible to repeatedly taking the locally validation-best batch (Cavusoglu et al., 2021, Maharana et al., 2023, Troshin et al., 20 Sep 2025, Arya et al., 21 May 2026, Tang et al., 26 Dec 2025).

Across these formulations, diversity-and-difficulty-aware sampling is best understood not as one algorithm but as a design principle: selection pressure is distributed between coverage of the data or output space and concentration on regions where the current model, decoder, or curriculum is weakest. The specific mathematical realization varies substantially, but the cited literature consistently treats the joint optimization of these two factors as more effective than either factor in isolation (Zhang et al., 14 Mar 2025, Maharana et al., 2023, Chen et al., 2023, Rao et al., 22 May 2025).