Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexity

Parallelized Inference Algorithms for Diffusion Models

Diffusion models have surged as the leading methodology for generative modeling in various domains, ranging from image generation to natural language processing. However, the computational cost associated with these models, particularly in the inference phase, is a bottleneck due to the extensive sequential iterations required. This paper proposes Parallelized Inference Algorithms for Diffusion Models (PIADM) that leverage parallel sampling techniques to mitigate this challenge. By introducing sub-linear time complexity solutions, the authors present the first implementation with provable sub-linear complexity regarding data dimension ( d ).

Key Contributions

1. Parallelized Inference Algorithms: The paper introduces parallelized inference algorithms for both Stochastic Differential Equations (SDE) and Probability Flow Ordinary Differential Equations (ODE) implementations of diffusion models (referred to as PIADM-SDE and PIADM-ODE, respectively).
2. Theoretical Guarantees: It provides rigorous convergence analysis of PIADM-SDE, demonstrating a poly-logarithmic time complexity (\widetilde{\mathcal{O}}(\log d)), and compatibility with the probability flow ODE implementation, achieving (\widetilde{\mathcal{O}}(\log (d \delta^{-2}))) time complexity. This significantly improves over previous state-of-the-art polynomial complexities.
3. Space Complexity Improvement: While PIADM-SDE maintains a (\widetilde{\mathcal{O}}(d²⁾⁾ space complexity, PIADM-ODE further improves the space complexity to (\widetilde{\mathcal{O}}(d^{3/2})).

Theoretical Underpinnings

The algorithms are rooted in the efficacy of the Picard iteration in solving nonlinear ODEs, which provides a scalable and efficient way to handle high-dimensional generative tasks. Each block of the time horizon is discretized, allowing parallel evaluations of the score function. The exponential integrator method and adaptive step sizes, particularly towards the end of the diffusion process, are critical to maintaining the accuracy and stability of the proposed methods.

Main Theorems

1. PIADM-SDE:

   • Assumptions: Relies on (L^2([0, t_N])) (\delta)-accuracy of the learned score, regular and normalized data distributions, and bounded (C¹⁾ norm with Lipschitz continuity for the learned NN-based scores.
   • Result: Achieves (\widetilde{\mathcal{O}}(\log (d \delta^{-2}))) approximate time complexity with (\widetilde{\mathcal{O}}(d² \delta^{-2})) space complexity.
   • Convergence: The error bound shows convergence to the true distribution (p_\eta) within (\delta)-accuracy.

2. PIADM-ODE:

   • Assumptions: Builds on (L^\infty([0, t_N])) (\delta)-accuracy of the learned score, the bounded (C¹⁾ norm of both true scores and learned scores, and the Lipschitz continuity of the true scores.
   • Result: Matches time complexity of (\widetilde{\mathcal{O}}(\log d)) and offers better space complexity (\widetilde{\mathcal{O}}(d^{3/2} \delta^{-1})) due to deterministic approximation.
   • Convergence and Correction Step: Introduces an additional corrector step using underdamped Langevin dynamics to refine the 2-Wasserstein distance bound to a Total Variation bound.

Implications and Future Directions

The innovations presented in this paper open up substantial practical and theoretical implications for AI:

1. Practical Implications: The proposed algorithms significantly reduce the computational cost and time for generating high-quality samples from diffusion models, making it feasible to deploy these models in real-time applications, especially in environments with modern GPU clusters.
2. Theoretical Implications: The introduction of provable sub-linear complexity algorithms marks a foundational step in the optimization of high-dimensional generative models. Future work could further refine these parallelization strategies, exploring their applications to other variational frameworks and potentially extending them to different types of neural network architectures.
3. Speculative Developments: The approach may inspire further research into hybrid models combining the benefits of SDEs and ODEs, developing more sophisticated corrector steps, and investigating new realms of data representations that can capitalize on the computational efficiencies introduced here.

Conclusion

The research outlined in this paper presents a substantial advancement in the field of generative modeling with diffusion models. By leveraging parallel sampling techniques and rigorous theoretical frameworks, the authors provide solutions that significantly enhance computational efficiency without compromising on the quality of the generated samples. This work sets the stage for further exploration and optimization in the burgeoning field of generative AI models.