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Provable Maximum Entropy Manifold Exploration via Diffusion Models
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Casts exploration as entropy maximization over the data manifold implicitly defined by a pre-trained diffusion model, and solves it via score-based mirror descent that sequentially fine-tunes the diffusion model with adjoint matching, with convergence proofs via mirror flows.
Problem
Generative models capture data distributions but do not explore beyond high-density regions; exploration (maximizing entropy over the approximate data manifold) is hard because it requires density estimation in high dimensions.
Approach
Uses the first variation of entropy (negative log density), whose gradient equals the negative score of the pre-trained diffusion model, so exploration reward can be optimized with first-order adjoint matching without density estimation; iterating this as mirror descent with KL regularization is shown to converge to the maximum-entropy distribution on the manifold.
Key results
- On the illustrative task, the fine-tuned model after K=4 mirror descent steps achieves significantly higher estimated entropy than the pre-trained model (pre-trained estimate -1916.47 vs. 564.72 for first iterate in cross-entropy table).
- In text-to-image (SD-1.5, prompt 'A creative architecture.'), FID between fine-tuned and pre-trained distributions rises to 10.25, 9.83, 19.15 for k=1,2,3 while CLIP stays near pre-trained level (22.27 vs 20.79, 20.88, 19.86).
- Vendi diversity increases from 1.53 (pre-trained) to 1.7 ± 0.08 after one iteration, with CLIP at 0.22 for both.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Fine-tune the model for 1 iteration | Stable Diffusion 1.5 with score-based exploration · Creative architecture | FID: 0.0 -> 10.25 | Helped |
| Fine-tune the model for 1 iteration | Stable Diffusion 1.5 with score-based exploration · Creative architecture | CLIP: 22.27 -> 20.79 | Hurt |
| Fine-tune the model for 1 iteration | Stable Diffusion 1.5 with score-based exploration · Creative architecture | Cross-entropy: -1916.47 -> 564.72 | Helped |
| Fine-tune the model for 2 iterations | Stable Diffusion 1.5 with score-based exploration · Creative architecture | FID: 0.0 -> 9.83 | Helped |
| Fine-tune the model for 2 iterations | Stable Diffusion 1.5 with score-based exploration · Creative architecture | CLIP: 22.27 -> 20.88 | Hurt |
| Fine-tune the model for 2 iterations | Stable Diffusion 1.5 with score-based exploration · Creative architecture | Cross-entropy: -1916.47 -> 482.81 | Helped |
| Fine-tune the model for 3 iterations | Stable Diffusion 1.5 with score-based exploration · Creative architecture | FID: 0.0 -> 19.15 | Helped |
| Fine-tune the model for 3 iterations | Stable Diffusion 1.5 with score-based exploration · Creative architecture | CLIP: 22.27 -> 19.86 | Hurt |
| Fine-tune the model for 3 iterations | Stable Diffusion 1.5 with score-based exploration · Creative architecture | Cross-entropy: -1916.47 -> 843.88 | Helped |
| Fine-tune the model for 1 iteration | Stable Diffusion 1.5 with score-based exploration · Creative architecture | Vendi: $1.53$ -> $1.7\pm 0.08$ | Helped |
| Fine-tune the model for 2 iterations | Stable Diffusion 1.5 with score-based exploration · Creative architecture | Vendi: $1.53$ -> $1.63 \pm 0.07$ | Helped |
| Fine-tune the model for 1 iteration | Stable Diffusion 1.5 with score-based exploration · Creative architecture | CLIP: $0.22$ -> $0.22 \pm 0.01$ | No clear effect |
Surprising findings
- CLIP score drops only modestly (22.27 to 19.86) while distributional distance grows, suggesting exploration preserves semantic faithfulness.
Limitations
- Entropy cannot be directly evaluated for high-dimensional image models; proxy metrics (FID, CLIP, cross-entropy) are used instead.
- Convergence theorem relies on idealized or stochastic-approximation assumptions (exact score, support compatibility, noise/bias conditions) that may not hold in practice.
- Theorem 1 one-step optimality assumes exact score estimation and exact optimization.
Experimental setup
Datasets: LAION-5B
Benchmarks: synthetic 2D unbalanced distribution
Metrics: Monte Carlo entropy estimate FID CLIP score cross-entropy in Inception-v3 feature space Vendi score
Baselines: pre-trained Stable Diffusion 1.5
Builds on: Adjoint Matching mirror descent denoising diffusion probabilistic models relative smoothness analysis mirror flows
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