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O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions
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abstract
Score-based diffusion models, which generate new data by learning to reverse a diffusion process that perturbs data from the target distribution into noise, have achieved remarkable success across various generative tasks. Despite their superior empirical performance, existing theoretical guarantees are often constrained by stringent assumptions or suboptimal convergence rates. In this paper, we establish a fast convergence theory for the denoising diffusion probabilistic model (DDPM), a widely used SDE-based sampler, under minimal assumptions. Our analysis shows that, provided $\ell_{2}$-accurate estimates of the score functions, the total variation distance between the target and generated distributions is upper bounded by $O(d/T)$ (ignoring logarithmic factors), where $d$ is the data dimensionality and $T$ is the number of steps. This result holds for any target distribution with finite first-order moment. Moreover, we show that with careful coefficient design, the convergence rate improves to $O(k/T)$, where $k$ is the intrinsic dimension of the target data distribution. This highlights the ability of DDPM to automatically adapt to unknown low-dimensional structures, a common feature of natural image distributions. These results are achieved through a novel set of analytical tools that provides a fine-grained characterization of how the error propagates at each step of the reverse process.
Forward citations
Cited by 5 Pith papers
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A Sharp KL-Convergence Analysis for Diffusion Models under Minimal Assumptions
A new analysis shows O~(d/epsilon) steps suffice for KL-close diffusion sampling under only L2 score error and finite second moment assumptions, improving the known O~(d/epsilon^2).
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Faster Diffusion Models via Higher-Order Approximation
A new higher-order ODE sampler for diffusion models is proven to reach ε total-variation accuracy with eO(d^{1+2/K}/ε^{1/K}) iterations under mild assumptions.
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Almost Linear Convergence under Minimal Score Assumptions: Quantized Transition Diffusion
QTD turns continuous data into binary codes and uses a Hamming-distance Markov chain with truncated uniformization to sample, provably reaching epsilon TV error with O(d ln^2(d/epsilon)) score evaluations.
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Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Models
A TV convergence bound O(d^{7/4} ε^{1/2} + d(dH)^p) is proved for p-th order (exponential) Runge-Kutta samplers of probability-flow ODEs under C² smoothness of the learned score.
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Non-asymptotic convergence bound of conditional diffusion models
CARD's generated conditional distribution is shown to converge in Wasserstein distance to the true conditional distribution, with a separate score-estimation error bound controlled by network resolution and distributi...
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