Non-asymptotic error bounds for probability flow ODEs under weak log-concavity

TL;DR

This paper addresses non-asymptotic convergence of probability-flow ODE samplers in score-based models under weak log-concavity. It derives $\mathcal{W}_2$ convergence bounds that decompose into initialization, discretization, and score-matching errors using an exponential integrator, and applies to general drift/diffusion settings, including Gaussian mixtures. A specialized Ornstein–Uhlenbeck analysis provides explicit rate expressions and practical hyperparameter guidelines, while a regime-shift analysis explains the time-varying log-concavity properties. The results show that the asymptotics of the bounds match those obtained under strong log-concavity, thereby broadening the scope of provable guarantees for diffusion-based samplers and guiding practitioners in parameter selection. Overall, the work bridges rigorous theory and practical deterministic sampling for more realistic data distributions.

Abstract

Score-based generative modeling, implemented through probability flow ODEs, has shown impressive results in numerous practical settings. However, most convergence guarantees rely on restrictive regularity assumptions on the target distribution -- such as strong log-concavity or bounded support. This work establishes non-asymptotic convergence bounds in the 2-Wasserstein distance for a general class of probability flow ODEs under considerably weaker assumptions: weak log-concavity and Lipschitz continuity of the score function. Our framework accommodates non-log-concave distributions, such as Gaussian mixtures, and explicitly accounts for initialization errors, score approximation errors, and effects of discretization via an exponential integrator scheme. Bridging a key theoretical challenge in diffusion-based generative modeling, our results extend convergence theory to more realistic data distributions and practical ODE solvers. We provide concrete guarantees for the efficiency and correctness of the sampling algorithm, complementing the empirical success of diffusion models with rigorous theory. Moreover, from a practical perspective, our explicit rates might be helpful in choosing hyperparameters, such as the step size in the discretization.

Figures (3)

• Figure 1: Plots corresponding to a Gaussian mixture model. See Example \ref{['ex:gmm']} for more details.
• Figure 2: Plots corresponding to a constructed probability density function that is sub-gaussian but not weakly log-concave. See Example \ref{['ex:non_weak_concave']} for more details.
• Figure 3: Plot of $K(t) = \alpha(t) - M(t), \, t\geq 0$ for different values of $\alpha_0, M_0$, in the OU case.

Theorems & Definitions (43)

• Definition 1: Weak convexity
• Remark 2: General $f_M(r)$
• Proposition 3: Propagation of weak log-concavity in time
• Proposition 4: Regime shifting
• Proposition 5: Propagation of Lipschitz continuity in time
• Theorem 6: Error bound for the OU process
• Theorem 7: Error bound for the probability flow ODE
• Proposition 8: Comparison to the strongly log-concave case
• Proposition 9: Initialization error
• Proposition 10: Discretization and propagated score matching error