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KL Convergence Guarantees for Score diffusion models under minimal data assumptions

Giovanni Conforti, Alain Durmus, Marta Gentiloni Silveri

TL;DR

The paper addresses the problem of providing quantitative KL convergence guarantees for score-based diffusion models without assuming Lipschitz regularity, by analyzing OU and kinetic OU forward processes with fixed-step discretization and score estimators under minimal data conditions. It develops a stochastic-control perspective, introducing a relative score process and adjoint equations to derive explicit, sharp KL bounds that depend only on the data's Fisher information with respect to a Gaussian reference, with separate treatments for absolute and relative score errors. The main contributions are direct KL bounds for both OU and kOU SGMs (without early stopping or Lipschitz assumptions), improvements via relative error assumptions, and step-size strategies (including exponential-then-linear schemes) that yield favorable scaling in dimension and information content. These results bridge theory and practice by offering practical guidance on discretization and score-learning under mild data conditions and provide a foundation for extending the approach to more general forward diffusions.

Abstract

Diffusion models are a new class of generative models that revolve around the estimation of the score function associated with a stochastic differential equation. Subsequent to its acquisition, the approximated score function is then harnessed to simulate the corresponding time-reversal process, ultimately enabling the generation of approximate data samples. Despite their evident practical significance these models carry, a notable challenge persists in the form of a lack of comprehensive quantitative results, especially in scenarios involving non-regular scores and estimators. In almost all reported bounds in Kullback Leibler (KL) divergence, it is assumed that either the score function or its approximation is Lipschitz uniformly in time. However, this condition is very restrictive in practice or appears to be difficult to establish. To circumvent this issue, previous works mainly focused on establishing convergence bounds in KL for an early stopped version of the diffusion model and a smoothed version of the data distribution, or assuming that the data distribution is supported on a compact manifold. These explorations have led to interesting bounds in either Wasserstein or Fortet-Mourier metrics. However, the question remains about the relevance of such early-stopping procedure or compactness conditions. In particular, if there exist a natural and mild condition ensuring explicit and sharp convergence bounds in KL. In this article, we tackle the aforementioned limitations by focusing on score diffusion models with fixed step size stemming from the Ornstein-Uhlenbeck semigroup and its kinetic counterpart. Our study provides a rigorous analysis, yielding simple, improved and sharp convergence bounds in KL applicable to any data distribution with finite Fisher information with respect to the standard Gaussian distribution.

KL Convergence Guarantees for Score diffusion models under minimal data assumptions

TL;DR

The paper addresses the problem of providing quantitative KL convergence guarantees for score-based diffusion models without assuming Lipschitz regularity, by analyzing OU and kinetic OU forward processes with fixed-step discretization and score estimators under minimal data conditions. It develops a stochastic-control perspective, introducing a relative score process and adjoint equations to derive explicit, sharp KL bounds that depend only on the data's Fisher information with respect to a Gaussian reference, with separate treatments for absolute and relative score errors. The main contributions are direct KL bounds for both OU and kOU SGMs (without early stopping or Lipschitz assumptions), improvements via relative error assumptions, and step-size strategies (including exponential-then-linear schemes) that yield favorable scaling in dimension and information content. These results bridge theory and practice by offering practical guidance on discretization and score-learning under mild data conditions and provide a foundation for extending the approach to more general forward diffusions.

Abstract

Diffusion models are a new class of generative models that revolve around the estimation of the score function associated with a stochastic differential equation. Subsequent to its acquisition, the approximated score function is then harnessed to simulate the corresponding time-reversal process, ultimately enabling the generation of approximate data samples. Despite their evident practical significance these models carry, a notable challenge persists in the form of a lack of comprehensive quantitative results, especially in scenarios involving non-regular scores and estimators. In almost all reported bounds in Kullback Leibler (KL) divergence, it is assumed that either the score function or its approximation is Lipschitz uniformly in time. However, this condition is very restrictive in practice or appears to be difficult to establish. To circumvent this issue, previous works mainly focused on establishing convergence bounds in KL for an early stopped version of the diffusion model and a smoothed version of the data distribution, or assuming that the data distribution is supported on a compact manifold. These explorations have led to interesting bounds in either Wasserstein or Fortet-Mourier metrics. However, the question remains about the relevance of such early-stopping procedure or compactness conditions. In particular, if there exist a natural and mild condition ensuring explicit and sharp convergence bounds in KL. In this article, we tackle the aforementioned limitations by focusing on score diffusion models with fixed step size stemming from the Ornstein-Uhlenbeck semigroup and its kinetic counterpart. Our study provides a rigorous analysis, yielding simple, improved and sharp convergence bounds in KL applicable to any data distribution with finite Fisher information with respect to the standard Gaussian distribution.
Paper Structure (24 sections, 17 theorems, 123 equations, 1 figure)

This paper contains 24 sections, 17 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Score generative models
  4. Convergence bounds for OU-based SGM
  5. Convergence bounds for the kOU-based SGM
  6. Related works and comparison with existing literature
  7. Contribution of this work
  8. Proofs
  9. OU case
  10. Proof of Theorem \ref{['theo_absolute_OU_just_fisher']} and \ref{['theo_relative_OU_just_fisher']}
  11. Proof of \ref{['theo_exponential_then_linear']}
  12. kOU case
  13. Proof of Theorem \ref{['theo_absolute_kOU']} and \ref{['theo_relative_kOU']}
  14. Technical results
  15. Proof of \ref{['mapm_smooth']}
  16. ...and 9 more sections

Key Result

Theorem 1

Let $T\geqslant1,h\leqslant1$ and assume ass:estimation-absolute-ass:hyp_on_mustar. Consider the EI scheme $(X^{\theta^{\star}}_t)_{t\in [0,T]}$ with constant step size $h>0$ defined by eq:SGM-OU-Based. Denoting for any $t \in [0,T]$ by $p^{\theta^{\star}}_t$ the distribution of $X^{\theta^{\star}}_ where $\mathbf{C}(T,\varepsilon) = T \varepsilon^2$. Moreover, the bound i also holds if we replace

Figures (1)

  • Figure 1: Example of sequences $(h_k)_{k\in\{1,\ldots,N\}}$ and $(t_k)_{k\in\{1,\ldots,N\}}$ for $T= 4+ 2c$, $c=0.15$ and $a = 1/3$.

Theorems & Definitions (31)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • proof
  • Theorem 4
  • Theorem 5
  • ...and 21 more