Diffusion Posterior Sampling is Computationally Intractable (2402.12727v1)

Abstract: Diffusion models are a remarkably effective way of learning and sampling from a distribution $p(x)$. In posterior sampling, one is also given a measurement model $p(y \mid x)$ and a measurement $y$, and would like to sample from $p(x \mid y)$. Posterior sampling is useful for tasks such as inpainting, super-resolution, and MRI reconstruction, so a number of recent works have given algorithms to heuristically approximate it; but none are known to converge to the correct distribution in polynomial time. In this paper we show that posterior sampling is \emph{computationally intractable}: under the most basic assumption in cryptography -- that one-way functions exist -- there are instances for which \emph{every} algorithm takes superpolynomial time, even though \emph{unconditional} sampling is provably fast. We also show that the exponential-time rejection sampling algorithm is essentially optimal under the stronger plausible assumption that there are one-way functions that take exponential time to invert.

• The paper establishes that diffusion model posterior sampling is computationally intractable under basic cryptographic assumptions.
• It demonstrates that any algorithm attempting posterior sampling requires superpolynomial time, contrasting with efficient unconditional sampling.
• The findings urge a reassessment of current strategies and motivate the search for new algorithms to overcome inherent computational limitations.

Computational Intractability of Diffusion Posterior Sampling Under Cryptographic Assumptions

Introduction

The analysis of diffusion processes in the context of generative models, specifically posterior sampling, is critical in understanding the boundaries of these models' applications. This exploration is pertinent to tasks like inpainting, super-resolution, and MRI reconstruction, where one aims to recover original sample distributions from noisy measurements. Recent advancements in diffusion models have demonstrated impressive capabilities in unconditional sampling from complex distributions, notably image distributions. However, the efficiency and theoretical guaranties of these models in posterior sampling remain largely unexplored. This paper addresses the computational feasibility of posterior sampling in diffusion models under standard cryptographic assumptions, presenting a nuanced perspective on the limitations inherent to these methods.

Theoretical Contributions

In a significant theoretical contribution, this research establishes that posterior sampling, despite its utility, is computationally intractable in certain scenarios under fundamental cryptographic assumptions. The crux of the argument rests on the existence of one-way functions—a cornerstone in cryptography. The paper demonstrates that for any algorithm, attempting posterior sampling in the proposed scenario requires superpolynomial time, contrasting sharply with the polynomial-time efficiency observed in unconditional sampling contexts. Moreover, it posits that even if approximations of smoothed scores are provided, achieving fast and robust posterior sampling remains elusive. This revelation aligns with the stronger assumption that if certain one-way functions necessitate exponential time to invert, posterior sampling's computational demands scale similarly, underscoring an intrinsic complexity that challenges current algorithmic approaches.

Practical Implications

From a practical standpoint, the results underscore a fundamental limitation in leveraging diffusion models for posterior sampling tasks. Given diffusion models' growing prominence in generating high-fidelity samples for a broad range of applications, recognizing these computational boundaries is crucial. This understanding prompts a reevaluation of current strategies and encourages the search for novel algorithms or theoretical frameworks that can accommodate or circumvent these limitations. The identification of distributional properties that could mitigate the computational intractability presents an intriguing avenue for future research.

Speculations on Future Directions

This work invites further scholarly pursuit in several directions. Principally, it opens up a dialogue on the nature of distributions for which posterior sampling might be feasible without encountering the prohibitive computational overhead identified. Investigating alternative algorithmic strategies or modified diffusion models that could offer more favorable computational properties for posterior sampling also presents an interesting challenge. Moreover, the exploration of practical approximations that, while not fully accurate, could provide sufficient efficacy for specific applications warrants attention. Further cryptographic analysis could also reveal more about the relationship between diffusion models' capabilities and foundational cryptographic principles.

Concluding Remarks

In summary, this research furnishes critical insights into the computational landscape of diffusion models, particularly highlighting a significant bottleneck in posterior sampling. By framing this limitation within the context of established cryptographic assumptions, it not only delineates the boundaries of current methodologies but also lays the groundwork for a nuanced understanding and future exploration of generative models. As the field continues to evolve, addressing the dichotomy between theoretical intractability and practical necessity will undoubtedly shape the trajectory of research and applications in artificial intelligence and machine learning.