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Provable Sample-Efficient Transfer Learning Conditional Diffusion Models via Representation Learning

Ziheng Cheng, Tianyu Xie, Shiyue Zhang, Cheng Zhang

TL;DR

The paper tackles the data inefficiency of conditional diffusion models by introducing a representation-learning framework that captures a shared low-dimensional conditioning space across tasks. It proves generalization and end-to-end distribution-estimation guarantees for transfer/meta-learning CDMs, leveraging deep ReLU networks to approximate both the score function and condition representations. The results show improved sample efficiency in few-shot and meta-learning regimes, with theoretical rates and practical experiments on conditioned diffusion and MNIST image restoration. This work lays a foundation for principled transfer learning in unsupervised diffusion-based generation, bridging representation learning with probabilistic modeling and offering insights for real-world applications with limited target data.

Abstract

While conditional diffusion models have achieved remarkable success in various applications, they require abundant data to train from scratch, which is often infeasible in practice. To address this issue, transfer learning has emerged as an essential paradigm in small data regimes. Despite its empirical success, the theoretical underpinnings of transfer learning conditional diffusion models remain unexplored. In this paper, we take the first step towards understanding the sample efficiency of transfer learning conditional diffusion models through the lens of representation learning. Inspired by practical training procedures, we assume that there exists a low-dimensional representation of conditions shared across all tasks. Our analysis shows that with a well-learned representation from source tasks, the samplecomplexity of target tasks can be reduced substantially. In addition, we investigate the practical implications of our theoretical results in several real-world applications of conditional diffusion models. Numerical experiments are also conducted to verify our results.

Provable Sample-Efficient Transfer Learning Conditional Diffusion Models via Representation Learning

TL;DR

The paper tackles the data inefficiency of conditional diffusion models by introducing a representation-learning framework that captures a shared low-dimensional conditioning space across tasks. It proves generalization and end-to-end distribution-estimation guarantees for transfer/meta-learning CDMs, leveraging deep ReLU networks to approximate both the score function and condition representations. The results show improved sample efficiency in few-shot and meta-learning regimes, with theoretical rates and practical experiments on conditioned diffusion and MNIST image restoration. This work lays a foundation for principled transfer learning in unsupervised diffusion-based generation, bridging representation learning with probabilistic modeling and offering insights for real-world applications with limited target data.

Abstract

While conditional diffusion models have achieved remarkable success in various applications, they require abundant data to train from scratch, which is often infeasible in practice. To address this issue, transfer learning has emerged as an essential paradigm in small data regimes. Despite its empirical success, the theoretical underpinnings of transfer learning conditional diffusion models remain unexplored. In this paper, we take the first step towards understanding the sample efficiency of transfer learning conditional diffusion models through the lens of representation learning. Inspired by practical training procedures, we assume that there exists a low-dimensional representation of conditions shared across all tasks. Our analysis shows that with a well-learned representation from source tasks, the samplecomplexity of target tasks can be reduced substantially. In addition, we investigate the practical implications of our theoretical results in several real-world applications of conditional diffusion models. Numerical experiments are also conducted to verify our results.

Paper Structure

This paper contains 43 sections, 31 theorems, 198 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Score Approximation and Distribution Estimation
  4. Transfer Learning and Meta-learning Theory in Supervised Learning
  5. Few-shot Fine-Tuning of Diffusion Models
  6. Preliminaries and Problem Setup
  7. Notations
  8. Conditional Diffusion Models
  9. Transfer Diffusion Models via Learning Representation
  10. Deep ReLU Neural Network Family
  11. Statistical Guarantees for Transferring Score Matching Error
  12. Tackling Lipschitz Continuity under Weaker Assumptions
  13. Results under Task Diversity: Sample-Efficient Transfer Learning
  14. Improving Sample Efficiency
  15. Results without Task Diversity: Meta-Learning Perspective
  16. ...and 28 more sections

Key Result

Lemma 3.1

Under Assumption asp:sub_gaussian, asp:low_dim, asp:lip, for any $w\in[0,1]^{d_y}$, denote the conditional score of forward process $\nabla_x\log p_t(x;w)$ by $f_*(x,w,t)$. There exist constants $C_X,C_X'$, such that for any $R>0$, the function $f_*(x,w,t)$ is $(C_X+C_X'R^2)$-Lipschitz in $x$, $(C_X

Theorems & Definitions (54)

  • Remark 1
  • Lemma 3.1
  • Definition 3.1: Task diversity
  • Proposition 3.2: Fine-tuning phase generalization
  • Proposition 3.3: Pre-training phase generalization
  • Theorem 3.4
  • Proposition 3.5: Generalization on meta distribution
  • Theorem 3.6
  • Theorem 4.1
  • Theorem 4.2: Transfer learning
  • ...and 44 more