Title: Prismatic Synthesis: Gradient-based Data Diversification Boosts Generalization in LLM Reasoning
Venue: NeurIPS 2025
Accepted: yes
Keywords: Large Language Models

Data Diversity

Data Quality

Synthetic Data

LLM Reasoning
Abstract: Data diversity is crucial for training a strong language model. Yet metrics of diversity often diverge from this goal, measuring variations in heuristic features—like n-grams or embeddings—that are detached from how the model actually performs on a target task. This motivates us to ask: *Can we redefine data diversity—beyond measuring variations in heuristic features—in a way that better predicts model generalization?* Through large-scale empirical analyses spanning over 300 training runs, carefully controlled for data scale and quality, we show that data diversity can be a strong predictor of generalization in LLM reasoning—as measured by average model performance on unseen out-of-distribution benchmarks. We introduce **G-Vendi**, a metric that quantifies diversity via the entropy of model-induced loss gradients. G-Vendi scales to million-sample datasets and yet consistently outperforms heuristic alternatives, achieving strong correlation ($\text{Spearman's } \rho \approx 0.9$) with out-of-distribution (OOD) performance across both natural language inference (NLI) and math reasoning tasks. Building on this insight, we present **Prismatic Synthesis**, a framework for generating diverse synthetic data by targeting underrepresented regions in gradient space. Experimental results show that Prismatic Synthesis consistently improves model performance as we scale synthetic data—not just on in-distribution test but across unseen, out-of-distribution benchmarks—significantly outperforming state-of-the-art models in both domains. For example, PrismMath-7B, our model distilled from a 32B LLM without human verification, outperforms R1-Distill-Qwen-7B—trained on proprietary data generated by 671B R1—on 6 out of 7 challenging math benchmarks.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes to measure the \"diversity\" of a dataset used to train a model by measuring the entropy of a proxy model's gradients on this data distribution. The paper shows that this measure outperforms other metrics (like embedding diversity), and shows how it can be used to guide synthetic data sampling for model training.\n\nSTRENGTHS AND WEAKNESSES:\n- The idea of using gradient diversity is a nice and conceptually well-motivated idea for measuring dataset diversity, and the use of random matrix projection to make this computation tractable enables this to be done in a reasonably scalable manner\n- Various aspects of the measure (e.g. the choice of proxy model, how it compares to various downstream metrics) are well analyzed\n- This metric seems to be better than other approaches (like embedding or perplexity-based metrics) in predicting utility\n- I thought the use of Diverse-{Reason, Persona} to disentangle what diversity from embedding vs gradient diversity looks like to be didactic\n- The paper is well-written and easy to follow.\n\nWeaknesses:\n\nFor the comparisons in Table 4, it would have been useful to compare the G-Vendi approach to performing data generation with other diversity metrics that were measured in prior sections (like embedding diversity, perplexity, etc). As it stands right now, it is hard to draw conclusions since the data generation is not controlled for between Prism-{NLI, Math} and the other comparisons.\n\nQUESTIONS:\n1. Is the gradient entropy measure useful only from \"base\" models, or is it a useful metric, say during training? E.g. what does the gradient entropy for a dataset look like using model iterates from fine-tuning?\n2. What is the relative cost of computing the gradient entropy? Given a batch of $bs \\times seq\\_len$ tokens, what is the cost of computing the gradient entropy vs computing the avg minibatch gradient?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

comments: SUMMARY:
This paper discussed measurement of data diversity and proposes a method to amplify it. First, it introduces G-Vendi, a metric that measures the diversity of a given dataset. Moreover, it proposes Prismatic Synthesis that can synthesize new data to expand the diversity. Experiments demonstrate the improvements.

STRENGTHS AND WEAKNESSES:
Strengths:
1. It explores the measurement of data diversity, and first proposes the metric (i.e., G-Vendi) that precisely measures the diversity of a given dataset. This could be significantly beneficial for collecting data and data postprocessing. 
2. Experiments are good and extensive, and the compared counterparts are sufficient.
3. The overall presentation and writing are clear. 

Weaknesses:
1. The comparison in experiments in Table 2 is sparse. The gap between DiverseReason and DiversePersona is too large to prove that G-Vendi is a sensitive metric. Maybe an intermediate dataset should be added in this comparison.
2. Experiments in Table 3 are based on small models (<= 1B). How about larger ones (>= 7B)?

QUESTIONS:
See the weakneeses.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates the role of data diversity in improving generalization for LLMs, especially in reasoning tasks. The authors introduce a new metric, G-Vendi, which quantifies data diversity based on the entropy of model-induced gradients. This measure is shown to be strongly correlated with OOD generalization performance even when using a small off-the-shelf proxy model. Building on this insight, the paper proposes Prismatic Synthesis, a gradient-space-driven synthetic data generation framework. This framework identifies underrepresented gradient clusters and selectively samples new data points that enrich the gradient space. Using this technique, the authors construct PrismMath and PrismNLI, synthetic datasets that outperform several state-of-the-art models on a range of challenging benchmarks.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n- Understanding what makes training data effective for reasoning is a fundamental question in LLM development\n- The emphasis on gradient-based representations offers a new lens for understanding data quality. Specifically, G-Vendi metric is theoretically motivated and practically effective.  Prismatic Synthesis introduces a simple yet effective loop to diversify data based on underrepresented gradient space regions.\n- PrismMath-7B outperforms models trained on data from larger generators, demonstrating real utility\n\n\n## Weakness \n\n- Iteration specification: It is not specified how many iterations are used in Prismatic Synthesis (line 248)\n\n- There may be fundamental limits to how well gradient entropy on training data can predict reasoning ability on unseen benchmarks, particularly when training and evaluation domains differ significantly. The strong correlation observed may be partly due to domain similarity. For instance, the math experiments use OpenR1-Math as the seed dataset and evaluate on similar math reasoning tasks. Can G-Vendi/Prismatic Synthesis's effectiveness extend to other domains like code reasoning, logical reasoning, etc?  The connection between gradient entropy and downstream task performance is not very clear.\n\n- There is limited discussion on failure modes or cases where G-Vendi might underperform\n\n- Prismatic Synthesis overhead: The iterative clustering and generation process likely incurs high computational costs compared to simpler data selection methods, but the paper provides no direct comparison of computational overhead between Prismatic Synthesis and baseline approaches.\n\n- G-Vendi scalability: Computing gradient entropy for large datasets (e.g., the 100k samples in Figure 2) may be computationally expensive. While the authors mention O(d²|D|) complexity, practical wall-clock time and memory requirements are not provided. \n\n\n- Dimensionality reduction: The choice of 1024 dimensions for random projection (line 93) appears arbitrary without justification.\n- Clustering parameters: Important hyperparameters like k=1% of data pool size and retaining the top 50% sparsest clusters lack justification \n\n- Scale constraints: Experiments use relatively small models (1B-7B parameters); behavior at larger scales is uncertain\n\nQUESTIONS:\nsee Weaknesses part\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-07"}