Unified Convergence Analysis for Score-Based Diffusion Models with Deterministic Samplers

• URL: http://arxiv.org/abs/2410.14237v1
• Date: Fri, 18 Oct 2024 07:37:36 GMT
• Title: Unified Convergence Analysis for Score-Based Diffusion Models with Deterministic Samplers
• Authors: Runjia Li, Qiwei Di, Quanquan Gu,
• Abstract summary: We introduce a unified convergence analysis framework for deterministic samplers. Our framework achieves iteration complexity of $tilde O(d2/epsilon)$. We also provide a detailed analysis of Denoising Implicit Diffusion Models (DDIM)-type samplers.
• Score: 49.1574468325115
• License:
• Abstract: Score-based diffusion models have emerged as powerful techniques for generating samples from high-dimensional data distributions. These models involve a two-phase process: first, injecting noise to transform the data distribution into a known prior distribution, and second, sampling to recover the original data distribution from noise. Among the various sampling methods, deterministic samplers stand out for their enhanced efficiency. However, analyzing these deterministic samplers presents unique challenges, as they preclude the use of established techniques such as Girsanov's theorem, which are only applicable to stochastic samplers. Furthermore, existing analysis for deterministic samplers usually focuses on specific examples, lacking a generalized approach for general forward processes and various deterministic samplers. Our paper addresses these limitations by introducing a unified convergence analysis framework. To demonstrate the power of our framework, we analyze the variance-preserving (VP) forward process with the exponential integrator (EI) scheme, achieving iteration complexity of $\tilde O(d^2/\epsilon)$. Additionally, we provide a detailed analysis of Denoising Diffusion Implicit Models (DDIM)-type samplers, which have been underexplored in previous research, achieving polynomial iteration complexity.

Related papers

• Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers [49.97755400231656]
  We present the first performance guarantee with explicit dimensional general score-mismatched diffusion samplers. We show that score mismatches result in an distributional bias between the target and sampling distributions, proportional to the accumulated mismatch between the target and training distributions. This result can be directly applied to zero-shot conditional samplers for any conditional model, irrespective of measurement noise.
  arXiv  Detail & Related papers  (2024-10-17T16:42:12Z)
• Convergence of Score-Based Discrete Diffusion Models: A Discrete-Time Analysis [56.442307356162864]
  We study the theoretical aspects of score-based discrete diffusion models under the Continuous Time Markov Chain (CTMC) framework. We introduce a discrete-time sampling algorithm in the general state space $[S]d$ that utilizes score estimators at predefined time points. Our convergence analysis employs a Girsanov-based method and establishes key properties of the discrete score function.
  arXiv  Detail & Related papers  (2024-10-03T09:07:13Z)
• HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative models [1.949927790632678]
  This paper builds on the log transform known as the Cole-Hopf transform in Brownian motion contexts. We develop a new algorithm, named the HJ-sampler, for inference for the inverse problem of a differential equation with given terminal observations.
  arXiv  Detail & Related papers  (2024-09-15T05:30:54Z)
• Variational Learning of Gaussian Process Latent Variable Models through Stochastic Gradient Annealed Importance Sampling [22.256068524699472]
  In this work, we propose an Annealed Importance Sampling (AIS) approach to address these issues. We combine the strengths of Sequential Monte Carlo samplers and VI to explore a wider range of posterior distributions and gradually approach the target distribution. Experimental results on both toy and image datasets demonstrate that our method outperforms state-of-the-art methods in terms of tighter variational bounds, higher log-likelihoods, and more robust convergence.
  arXiv  Detail & Related papers  (2024-08-13T08:09:05Z)
• Training Implicit Generative Models via an Invariant Statistical Loss [3.139474253994318]
  Implicit generative models have the capability to learn arbitrary complex data distributions. On the downside, training requires telling apart real data from artificially-generated ones using adversarial discriminators. We develop a discriminator-free method for training one-dimensional (1D) generative implicit models.
  arXiv  Detail & Related papers  (2024-02-26T09:32:28Z)
• Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers [90.45898746733397]
  We develop a framework for non-asymptotic analysis of deterministic samplers used for diffusion generative modeling. We show that one step along the probability flow ODE can be expressed as two steps: 1) a restoration step that runs ascent on the conditional log-likelihood at some infinitesimally previous time, and 2) a degradation step that runs the forward process using noise pointing back towards the current gradient.
  arXiv  Detail & Related papers  (2023-03-06T18:59:19Z)
• Sampling from Arbitrary Functions via PSD Models [55.41644538483948]
  We take a two-step approach by first modeling the probability distribution and then sampling from that model. We show that these models can approximate a large class of densities concisely using few evaluations, and present a simple algorithm to effectively sample from these models.
  arXiv  Detail & Related papers  (2021-10-20T12:25:22Z)
• Oops I Took A Gradient: Scalable Sampling for Discrete Distributions [53.3142984019796]
  We show that this approach outperforms generic samplers in a number of difficult settings. We also demonstrate the use of our improved sampler for training deep energy-based models on high dimensional discrete data.
  arXiv  Detail & Related papers  (2021-02-08T20:08:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.