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A Theory for Conditional Generative Modeling on Multiple Data Sources
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Gives a bracketing-number-based average-TV error bound for conditional MLE and shows multi-source conditional generative training can beat single-source training when source distributions share parameters, instantiated for conditional Gaussians, ARMs and EBMs.
Problem
Theoretical understanding of whether training one conditional generative model on multiple data sources beats training separate single-source models, measured by distribution estimation error.
Approach
Extends classical MLE bracketing-number bounds to the conditional setting via one-sided upper brackets and average TV error; multi-source training shares parameters, reducing the bracketing number of the distribution space, yielding a provably sharper bound with ratio sqrt(1-((K-1)/K)*beta_sim).
Key results
- On ILSVRC2012 class-conditional diffusion, multi-source training achieved lower average FID than single-source in all 8 settings (e.g., N=500, Sim=2, K=10: 28.75 multi vs 30.54 single).
- Simulations on conditional Gaussians show empirical average TV error of multi-source training consistently below single-source, with scaling matching the O~(sqrt(K)) and O~(1/sqrt(n)) bounds.
- Theory gives multi-source error O~(sqrt(((K-1)d1+d)/n)) versus single-source O~(sqrt(Kd/n)) for Gaussians; ratio sqrt(1-((K-1)/K) beta_sim).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train one conditional diffusion model jointly across sources instead of separate models per source | EDM2 class-conditional diffusion models · ILSVRC2012, N=500, Sim=1, K=3 | Avg. FID: 30.03 -> 29.94 | Helped |
| Train one conditional diffusion model jointly across sources instead of separate models per source | EDM2 class-conditional diffusion models · ILSVRC2012, N=500, Sim=1, K=10 | Avg. FID: 30.18 -> 29.28 | Helped |
| Train one conditional diffusion model jointly across sources instead of separate models per source | EDM2 class-conditional diffusion models · ILSVRC2012, N=500, Sim=2, K=3 | Avg. FID: 32.69 -> 30.69 | Helped |
| Train one conditional diffusion model jointly across sources instead of separate models per source | EDM2 class-conditional diffusion models · ILSVRC2012, N=500, Sim=2, K=10 | Avg. FID: 30.54 -> 28.75 | Helped |
| Train one conditional diffusion model jointly across sources instead of separate models per source | EDM2 class-conditional diffusion models · ILSVRC2012, N=1000, Sim=1, K=3 | Avg. FID: 28.01 -> 26.41 | Helped |
| Train one conditional diffusion model jointly across sources instead of separate models per source | EDM2 class-conditional diffusion models · ILSVRC2012, N=1000, Sim=1, K=10 | Avg. FID: 27.49 -> 25.84 | Helped |
| Train one conditional diffusion model jointly across sources instead of separate models per source | EDM2 class-conditional diffusion models · ILSVRC2012, N=1000, Sim=2, K=3 | Avg. FID: 30.58 -> 29.35 | Helped |
| Train one conditional diffusion model jointly across sources instead of separate models per source | EDM2 class-conditional diffusion models · ILSVRC2012, N=1000, Sim=2, K=10 | Avg. FID: 29.01 -> 27.81 | Helped |
| Increase the number of jointly trained sources | Multi-source EDM2 class-conditional diffusion models · ILSVRC2012, fixed Sim and N | Relative Avg. FID advantage | Helped |
| Increase similarity among jointly trained sources | Multi-source EDM2 class-conditional diffusion models · ILSVRC2012, fixed K and N | Relative Avg. FID advantage | Helped |
| Use multi-source rather than single-source training | Conditional Gaussian estimation · Gaussian simulations across tested K, n, and similarity settings | Empirical average TV error | Helped |
Limitations
- Assumes realizable setting where true parameters lie in the model class and the source label distribution is known.
- Theory uses MLP-based ARM/EBM with specific architectures; language models lack explicit conditions in practice.
- Real-world experiments are limited to a few ImageNet class subsets with two manually designed similarity levels.
Experimental setup
Datasets: ILSVRC2012 ImageNet
Benchmarks: ILSVRC2012 subsets (256x256)
Metrics: average TV error (Monte Carlo estimate) average FID
Baselines: single-source training
Builds on: classical MLE error bounds via empirical process theory EDM2 Gaussian bracketing number analysis (Lemma C.5)
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