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Diffusion Posterior Sampling is Computationally Intractable

Shivam Gupta, Ajil Jalal, Aditya Parulekar, Eric Price, Zhiyang Xun

TL;DR

This work proves that posterior sampling for diffusion models under a linear measurement model is computationally intractable in general, assuming the existence of one-way functions. It contrasts efficient unconditional sampling, which diffusion models can achieve for well-modeled distributions, with hardness results for posterior inference, deriving both superpolynomial and exponential-time lower bounds. The authors develop a well-modeled hard instance $\widetilde{g}$ and show that any $(\tfrac{1}{10},\tfrac{1}{10})$-posterior sampler would imply inverting a one-way function, while also showing that the associated smoothed-score can be represented by a polynomial-size ReLU network. They further provide an upper-bound analysis via rejection sampling, illustrating that, under exponential-time inverters, rejection sampling is near-optimal. Overall, the paper delineates the limitations of general-purpose posterior sampling and motivates distributional assumptions or new algorithms tailored to specific task priors.

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 computationally intractable: under the most basic assumption in cryptography -- that one-way functions exist -- there are instances for which every algorithm takes superpolynomial time, even though 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.

Diffusion Posterior Sampling is Computationally Intractable

TL;DR

This work proves that posterior sampling for diffusion models under a linear measurement model is computationally intractable in general, assuming the existence of one-way functions. It contrasts efficient unconditional sampling, which diffusion models can achieve for well-modeled distributions, with hardness results for posterior inference, deriving both superpolynomial and exponential-time lower bounds. The authors develop a well-modeled hard instance and show that any -posterior sampler would imply inverting a one-way function, while also showing that the associated smoothed-score can be represented by a polynomial-size ReLU network. They further provide an upper-bound analysis via rejection sampling, illustrating that, under exponential-time inverters, rejection sampling is near-optimal. Overall, the paper delineates the limitations of general-purpose posterior sampling and motivates distributional assumptions or new algorithms tailored to specific task priors.

Abstract

Diffusion models are a remarkably effective way of learning and sampling from a distribution . In posterior sampling, one is also given a measurement model and a measurement , and would like to sample from . 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 computationally intractable: under the most basic assumption in cryptography -- that one-way functions exist -- there are instances for which every algorithm takes superpolynomial time, even though 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.
Paper Structure (49 sections, 56 theorems, 217 equations, 3 figures, 1 algorithm)

This paper contains 49 sections, 56 theorems, 217 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Precise statements.
  3. Related Work
  4. Posterior Score Approximation.
  5. Replacement Method.
  6. Particle Filtering.
  7. Proof Overview -- Lower Bound
  8. An initial attempt.
  9. Lower Bound Instance
  10. Posterior Sampling Implies Inversion
  11. ReLU Approximation of Lower Bound Score
  12. Notation.
  13. Strategy.
  14. ReLU Approximation for Score of Product Distribution
  15. ReLU Approximation for Large $\sigma$
  16. ...and 34 more sections

Key Result

Theorem 1.4

For an $O(C)$-well-modeled distribution $p$, the discretized reverse diffusion process with approximate scores gives a $\left(\frac{1}{d^{C}}, \frac{1}{d^{C}} \right)$-unconditional sampler (as defined in Definition def:unconditional_sampler) for any constant $C > 0$ in $\mathop{\mathrm{poly}}\nolim

Figures (3)

  • Figure 1: The distribution of each coordinate in $g_s$, has independent coordinates. For any seed $s \in \{\pm 1\}^d$, the first $d$ bits are normal distributions whose mean is specified by $s_i$, and the last $d'$ bits are a discretized standard normal where the discretization is specified by $f(s)_j$. The full distribution $g$ is a mixture over all seeds $s$ of $g_s$.
  • Figure 2: Piecewise-Linear Approximations of Score $s_\sigma$
  • Figure 3: $\psi_1$ and $\psi_{-1}$ are discretized Gaussians with discretization width $\varepsilon$ and phase $0$ and $\varepsilon/2$ respectively. If we convolve with $\mathcal{N}(0, \sigma^2)$, we get a distribution close to Gaussian when $\sigma \ge \varepsilon$ for each of $\psi_1, \psi_{-1}$.

Theorems & Definitions (110)

  • Definition 1.1: $C$-Well-Modeled Distribution
  • Definition 1.2: $(\tau, \delta)$-Close Distribution
  • Definition 1.3: $(\tau, \delta)$-Unconditional Sampler
  • Theorem 1.4: Unconditional Sampling for Well-Modeled Distributions
  • Definition 1.5: $(\tau, \delta)$-Posterior Sampler
  • Definition 1.6: Linear Measurement Model
  • Theorem 1.7: Upper Bound
  • Theorem 1.8: Lower Bound
  • Theorem 1.9: Lower Bound: Exponential Hardness
  • Remark 1.10
  • ...and 100 more