# A Theory for Conditional Generative Modeling on Multiple Data Sources

arXiv: https://arxiv.org/abs/2502.14583 · HTML: https://mlsanity.com/paper/2502.14583/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 63 references that are in the map.
- First posted on arXiv 20 Feb 2025.

## Builds on (most cited first, 30 of 46)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [PixArt-α: Fast Training of Diffusion Transformer for Photorealistic Text-to-Image Synthesis](https://mlsanity.com/paper/2310.00426/) (2023)
- [MiniCPM: Unveiling the Potential of Small Language Models with Scalable Training Strategies](https://mlsanity.com/paper/2404.06395/) (2024)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [How multilingual is Multilingual BERT?](https://mlsanity.com/paper/1906.01502/) (2019)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Analyzing and Improving the Training Dynamics of Diffusion Models](https://mlsanity.com/paper/2312.02696/) (2023)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Physics of Language Models: Part 3.1, Knowledge Storage and Extraction](https://mlsanity.com/paper/2309.14316/) (2023)
- [The Benefit of Multitask Representation Learning](https://mlsanity.com/paper/1505.06279/) (2015)
- [On the Theory of Transfer Learning: The Importance of Task Diversity](https://mlsanity.com/paper/2006.11650/) (2020)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [EGSDE: Unpaired Image-to-Image Translation via Energy-Guided Stochastic Differential Equations](https://mlsanity.com/paper/2207.06635/) (2022)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [Physics of Language Models: Part 3.3, Knowledge Capacity Scaling Laws](https://mlsanity.com/paper/2404.05405/) (2024)
- [Quality Not Quantity: On the Interaction between Dataset Design and Robustness of CLIP](https://mlsanity.com/paper/2208.05516/) (2022)
- [Neural Autoregressive Distribution Estimation](https://mlsanity.com/paper/1605.02226/) (2016)
- [Universality of empirical risk minimization](https://mlsanity.com/paper/2202.08832/) (2022)
- [Provable Guarantees for Gradient-Based Meta-Learning](https://mlsanity.com/paper/1902.10644/) (2019)
- [Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory](https://mlsanity.com/paper/2403.11968/) (2024)

## Built on by (most cited first, 1 of 1)

- [Scaling Diffusion Transformers Efficiently via μP](https://mlsanity.com/paper/2505.15270/) (2025)

## Abstract (arXiv)

The success of large generative models has driven a paradigm shift, leveraging massive multi-source data to enhance model capabilities. However, the interaction among these sources remains theoretically underexplored. This paper takes the first step toward a rigorous analysis of multi-source training in conditional generative modeling, where each condition represents a distinct data source. Specifically, we establish a general distribution estimation error bound in average total variation distance for conditional maximum likelihood estimation based on the bracketing number. Our result shows that when source distributions share certain similarities and the model is expressive enough, multi-source training guarantees a sharper bound than single-source training. We further instantiate the general theory on conditional Gaussian estimation and deep generative models including autoregressive and flexible energy-based models, by characterizing their bracketing numbers. The results highlight that the number of sources and similarity among source distributions improve the advantage of multi-source training. Simulations and real-world experiments are conducted to validate the theory, with code available at: https://github.com/ML-GSAI/Multi-Source-GM.