Do Large Language Models Capture the Diversity in their Training Data?

Large language models are trained to model conditional distributions over text, yet it remains inadequately understood whether they capture the full diversity of plausible outputs present in their training data. We study this question through an information-theoretic lens by comparing the conditional entropy of model-generated outputs with that of the corresponding training data. Given paired input-output samples, we use conditional entropy and its matrix-based analogue based on von Neumann entropy to measure output variability beyond what is explained by the conditioning input, without requiring multiple reference outputs for the same prompt. Across LLM families with publicly available training data, including OLMo, Pythia, and GPT-Neo, we consistently find that model-generated outputs exhibit lower conditional entropy than their training data, across different model scales, sequence lengths, and decoding strategies. We observe a similar conditional diversity gap beyond language modeling, including class-conditioned ImageNet generators and text-conditioned models trained on MS-COCO. To address this gap, we propose a post-hoc correction mechanism that generates multiple outputs for each input and reweights them through a matrix-entropy projection, increasing conditional diversity while remaining close to the original model distribution. We prove the concavity of the matrix-based conditional entropy functional, which makes the resulting entropy-constrained projection a convex optimization problem, and develop a scalable mirror-descent algorithm for its implementation. Our results reveal a systematic conditional diversity gap between modern generative models and their training data, and provide an information-theoretic framework for measuring and mitigating this gap.

Creative Generation via Multi-Agent Debate: Does Debate Suppress Diversity?

Creative generation tasks, such as narrative writing and scientific ideation, demand both high-quality outputs and distinct responses across independent runs to maximize exploration. Multi-Agent Debate (MAD) has shown strong quality gains on factual and reasoning tasks, making it a natural candidate for creative generation. However, we find its convergence-driven design actively suppresses output diversity across independent runs, creating an inherent trade-off with creative tasks. We theoretically show that preserving diversity among agents within each debate session is a necessary condition for achieving diverse outputs across independent runs. Building on this finding, we propose Creative-MAD, which introduces two synergistic interventions to sustain agent divergence. Specifically, Cognitive Lens Assignment counters identity drift by anchoring each agent to a distinct and persistent cognitive mode, while Embedding-based Peer Selection counters majority pull by limiting each agent's context to its most semantically distant peers. Experiments across four creative benchmarks demonstrate that Creative-MAD significantly enhances both lexical and semantic diversity while maintaining MAD's output quality.

Aligned but Flattened: Analyzing the Trade-off between Cultural Alignment and Diversity in LLMs

Cultural fine-tuning has become the de facto paradigm for building culture-aware large language models (LLMs), yet existing optimization exclusively for alignment scores provides an incomplete portrait of cultural fidelity by systematically obscuring inherent cultural diversity. This unidimensional evaluation lens prompts a fundamental question: do models genuinely perceive distinct cultural nuances, or do they merely memorize dominant cultural values? To address this, we propose a synergistic evaluation framework that jointly formalizes cultural alignment and diversity. Through extensive benchmarking of six mainstream LLMs on the World Values Survey, this framework uncovers a systematic and critical trade-off: the pursuit of cultural alignment consistently incurs an acute expense of diversity, leading to severe "cultural flattening." Investigating this behavioral shift, we demonstrate that these superficial alignment gains stem from models artificially anchoring to dominant majorities, converging onto a monolithic response pattern that wipes out the heterogeneous distributions inherent to human groups. Crucially, our mechanistic analysis suggests that this diversity collapse is not merely a behavioral anomaly but more likely a structural consequence of the low-rank bias inherent in neural network optimization. Therefore, our findings expose the limitations of current post-training paradigms and call for a shift toward alignment objectives that preserve cross-cultural pluralism.

Influence-Directed Distillation: Solving the Diversity Bottleneck in Sampled-Token On-Policy Distillation

Sampled-token on-policy distillation (OPD) efficiently transfers capabilities from teacher to student using student-generated tokens, requiring teacher probabilities only for sampled tokens. Yet it frequently suffers from diversity distillation failure: the student's pass@1 improves while its pass@$k$ plateaus, failing to inherit the teacher's diversity. To explain this, we introduce First-Order Local Entropy Influence, a signed first-order proxy that decouples each update's entropy effect into the teacher--student log-probability gap and the student's local probability structure, and empirically links entropy contraction to negative-influence positions. Motivated by this, we propose Influence-Directed Adaptive On-Policy Distillation (IDA-OPD): rather than relying on costly full-vocabulary Forward-KL objectives, it preserves entropy-expanding updates while replacing entropy-contracting ones with divergence-adaptive advantage shrinkage, using only the teacher's sampled-token log-probability. Experiments on reasoning-oriented distillation show IDA-OPD consistently improves pass@$k$, inheriting the teacher's diversity through distillation, matches the strongest teacher-informed methods at strictly lower cost, and broadly maintains vanilla OPD's pass@1, all without full-vocabulary teacher information.

XTC: Head-Aware Sampling by Excluding Top Choices

Standard decoding rules for autoregressive language models promote diversity by rescaling the full next-token distribution or truncating its low-probability tail. These strategies overlook a common regime of open-ended generation in which several continuations are plausible but too much probability mass remains concentrated on the most generic choice. We introduce XTC (Exclude Top Choices), a lightweight head-aware decoding operator that targets this regime directly. XTC identifies tokens whose probabilities exceed an absolute plausibility threshold $τ$: when at least two qualify, it removes the dominant eligible choices with probability $ρ$ and retains only the weakest plausible alternative before renormalization. Across 60 experiments on Gemma 3 27B Q4, Gemma 3 12B Q6, and DeepSeek R1 14B Q6, with scaling validation on Llama 3.3 70B Q4, XTC improves the diversity-repetition Pareto frontier. On creative generation, Distinct-2 increases by 11--15% and repeat trigrams decrease by 27--47% across the four models. Combined with temperature scaling, gains reach 38% in Distinct-2 and 71% in repeat-trigram reduction over baseline. A blinded Amazon Mechanical Turk study with 150 Master raters yields a 62.3% creativity preference for XTC ($p<10^{-4}$) without reduced fluency, while a GPT-4o control judge reproduces the Anthropic-judge direction on every measure. On IFEval with Llama 3.3 70B Q4, XTC preserves prompt-level strict accuracy within 1.7 percentage points of baseline while recovering most of the diversity gain; a temperature setting matched on Distinct-2 reduces IFEval by 8.8 points. The effect is additive with temperature and repetition penalties, robust across quantization levels and model families, and consistent across twelve prompt genres. XTC has been adopted by llama.cpp, ExLlamaV2, and text-generation-webui.

Imaginative Generative AI: Crossing the Entropy Wall into Worlds Beyond Imitation

Generative AI models are primarily designed to imitate the data distribution, an objective that neither corrects diversity lost by a learned generator nor defines how generation should extend beyond the diversity of the data itself. We introduce Imaginative Generative AI (IGA), a framework that makes diversity part of the target-distribution design problem: among distributions close to a reference, IGA selects one whose spectral diversity reaches a prescribed level. Diversity is measured by the von Neumann entropy of the generated distribution's kernel covariance operator in a fixed representation space, providing a reference-free representation-guided measure of how broadly probability mass occupies embedding directions. The spectral entropy of the population data distribution defines an Entropy Wall. Below the wall, IGA performs diversity repair, recovering variation that a learned generator has lost while remaining within the diversity level of the data. Beyond the wall, the data distribution itself becomes infeasible, and IGA deliberately departs from it to produce distributions with greater representation-relative spectral diversity, an operational notion of imaginative generation. These regimes form a single regularization path from imitation to imagination and define an i.i.d. target distribution at each prescribed diversity level. We develop the theory of this entropy-constrained projection and show that, under a KL anchor to a pretrained generator, the optimum satisfies a self-consistent exponential-tilt relation. This characterization leads to IGA Guidance, a retraining-free inference-time method for score-based and diffusion models, including DDPM and DDIM samplers. Experiments on synthetic and vision benchmarks demonstrate diversity repair below the Entropy Wall and controlled spectral extrapolation beyond it.

CreativeInstruct: Scalably Teaching LLMs to Balance Quality, Creativity, and Diversity

While post-training improves the capabilities of large language models (LLMs), it generally lowers their output diversity and creativity, negatively impacting tasks that explicitly require creativity (e.g., story generation) as well as those that require it implicitly, e.g., reinforcement learning (RL). We instead propose CreativeInstruct, a scalable instruction-tuning method that teaches LLMs to balance creative, base-model-like generations with the quality of post-trained models, by learning to inject special [StartCreativity] spans that bias generation toward creativity. Furthermore, we introduce a structural diversity metric based on graph edit distance, which captures narrative level variation missed by purely lexical and semantic metrics. On narrative generation, CreativeInstruct matches or exceeds the diversity of both multi-model baselines and distilled variants of their outputs, without sacrificing quality or requiring multiple models at inference time. These results are mirrored in our human evaluation, where we find that annotators rate CreativeInstruct generations as more creative than the post-trained LLMs' generations in 70.3% of cases. We also show the benefits of creative models as a substrate for RL: GRPO applied to a CreativeInstruct checkpoint improves by ~4% on AMC and ~5% points on MATH over the same training applied to the post-trained checkpoint.

Quantifying Diversity of Thought: A Predictive Law of Weighted LLM Ensemble Lift

This paper provides an experimentally verified formal law for calculating the uplift that diversity of thought provides in Large Language Model (LLM) ensembles. From first principles, we derive an exact decomposition of LLM ensemble lift into rescue and damage masses, which yields a compact heuristic for calculating uplift. From this we extract the metrics which predict ensemble performance: an accuracy-adjusted correctness correlation, $φ_{\mathrm{adj}}$, together with the accuracy gap and collective accuracy of the pair. We test the law on 767,520 inferences from ten open-weight models over two graduate-level science benchmarks, together with a novel agentic cybersecurity benchmark in which each model conducts digital-forensics investigations by multi-turn tool use in a network-isolated sandbox (23,520 graded trials including abstentions); all votes are released openly. Calibrated once on SuperGPQA at a 40:60 vote split, the heuristic predicts lift on the calibration set with Spearman's $ρ=0.84$ and, with its coefficients frozen, transfers to two datasets never used in calibration ($ρ=0.51$ on GPQA Diamond and $0.84$ on the forensic tasks), whilst the measured swap mass tracks realised lift with $R^2\ge 0.96$ throughout. Raw $φ$ has almost no predictive power ($R^2\le 0.09$ throughout); the accuracy-adjusted $φ_{\mathrm{adj}}$ is markedly superior ($R^2=0.67$ on SuperGPQA), and the heuristic combining these metrics is the most stable pre-pooling predictor across the three datasets.

Exploring More to Solve More: Boosting Diversity in Text Diffusion Models via Entropy-Based Guidance

Although diffusion models have revolutionized continuous domains like image synthesis through high quality generations and controllable guidance mechanisms, bringing this controllability to the discrete, sequential nature of text remains an open challenge. Meanwhile, current sampling strategies and guidance methods adjust token likelihoods without capturing the broader semantic landscape, leading to a suboptimal balance between fidelity and diversity. In this work, we introduce a novel training-free Semantic-Aware Kernel Entropy (SAKE) guidance method. Our method computes the order-2 Rényi entropy over a kernel Gram matrix that captures both cross-token semantic interactions and relative token positions. By linearizing this objective in the embedding space, we derive a tractable guidance signal that dynamically adjusts the sampling distribution, flattening it to encourage exploration during redundancy and sharpening it for fidelity when diverse. Empirical experiments demonstrate that our approach achieves a superior Pareto frontier between fidelity and diversity, and improves multi-sample performance on reasoning-intensive tasks, such as code and mathematics generation, compared to temperature scaling and discrete guidance baselines.

CuraWeb: Joint Optimization of Quality, Redundancy, and Diversity for Web-Scale Pretraining Data

Open-web corpora curated via highly selective filters, such as FineWeb-Edu and DCLM, constitute the core of LLM pretraining data and have significantly advanced LLM performance. However, these pipelines typically rely on singular optimization objectives, which inevitably narrows distributional diversity and marginalizes long-tail knowledge, thereby restricting data coverage and underutilizing the vast potential of the open web. To address this limitation, we propose a novel curation paradigm that shifts from linear pruning to the joint optimization of quality, redundancy, and diversity. This framework synergizes dual-track cleaning (rule-based and model-driven) with hybrid deduplication (n-gram and semantic), while employing a multi-objective sampler to balance informational quality with distributional breadth. Applying this framework to Common Crawl, we construct CuraWeb, a 2T-token English corpus. Unlike existing resources, CuraWeb establishes an industrial-grade standard for data curation by recovering a more holistic data distribution with enhanced diversity and minimal redundancy, achieving broader coverage of long-tail knowledge across diverse domains. Experimental evaluations at the 3B scale demonstrate that CuraWeb significantly outperforms state-of-the-art baselines, yielding an average performance gain of 1.8\% across a wide range of benchmarks, particularly in knowledge-intensive and reasoning tasks.

Breaking the Likelihood Trap: Variance-Calibrated Modulation for Large Language Model Decoding

In open-ended generation, LLMs frequently fall into the "likelihood trap", characterized by repetitive degeneration and vocabulary dullness, resulting in a discrepancy between machine-generated and human-written text. While post-hoc tail truncation (e.g., Top-p, Min-p) avoids sampling from the unreliable tail, it can misalign generation with human lexical preferences by over-sampling from the uncalibrated head; fixed scalar repetition penalties, in turn, ignore how the scale of the logit distribution varies across inference steps, which can disrupt semantic coherence. To address both shortcomings, we propose Variance-Calibrated Modulation (VCM), a training-free pre-decoding intervention. VCM directly reshapes the probability distribution prior to truncation via two dynamic mechanisms: (1) Contextual Searchlight via PMI, which naturally suppresses global stopwords and elevates context-evoked tokens, and (2) Adaptive Self-Debiasing, which utilizes real-time logit standard deviation to provide scale-invariant penalization. In experiments across open-ended generation, factual QA, and mathematical reasoning, we show that VCM consistently mitigates the likelihood trap. With negligible computational overhead, VCM integrates with existing decoding strategies, improving diversity and coherence and, particularly at higher decoding temperatures, reasoning accuracy. Our code is publicly available on GitHub: https://github.com/AetherDing/VCM

Spokes: Optimizing for Diverse Pretraining Data Selection

Diversity plays a critical role in data selection, improving performance under fixed data budgets by reducing redundancy and repetition. However, optimizing for diversity is inherently challenging, as it is a set-level property that depends on interactions between data points rather than individual examples. As a result, existing approaches typically rely on proxies or approximations, which often fail to ensure sufficiently diverse subsets. In this work, we directly optimize diversity by introducing a probabilistic diversification framework based on the G-Vendi score, optimized via exponentiated gradient descent. Our method produces subsets that are substantially more diverse than those obtained via random sampling, achieving a +489 increase in G-Vendi score on a 500k-sample subset. We evaluate our approach on FineWeb and DCLM, where it consistently outperforms existing methods. Notably, SPOKES (diversity-only) improves average downstream performance by +0.4 and +0.5 points over random sampling on DCLM and FineWeb, respectively. More importantly, jointly optimizing for both quality and diversity yields the strongest results: SPOKES achieves gains of +1.5 and +1.4 points on DCLM and FineWeb, outperforming all baselines, including semantic deduplication and quality filtering.

Beyond Temperature: Hyperfitting as a Late-Stage Geometric Expansion

Recent work has identified a counterintuitive phenomenon termed "Hyperfitting", where fine-tuning Large Language Models (LLMs) to near-zero training loss on small datasets surprisingly enhances open-ended generation quality and mitigates repetition in greedy decoding. While effective, the underlying mechanism remains poorly understood, with the extremely low-entropy output distributions suggesting a potential equivalence to simple temperature scaling. In this work, we demonstrate that this phenomenon is fundamentally distinct from distribution sharpening; entropy-matched control experiments reveal that temperature scaling fails to replicate the diversity gains of hyperfitting. Furthermore, we falsify the hypothesis of static vocabulary reweighting, showing through ablation studies that hyperfitting relies on a dynamic, context-dependent rank reordering mechanism. Layer-wise analysis localizes this effect to a "Terminal Expansion" in the final transformer block, where a substantial geometric expansion of the feature space (Delta Dim approx +80.8) facilitates the promotion of deep-tail tokens. Additionally, we introduce Late-Stage LoRA, a targeted fine-tuning strategy that updates only the final 5 layers, yielding robust generation with minimal parameter updates