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O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions
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Proves an O(d/T) total-variation convergence bound for the DDPM sampler under only finite-first-moment data and l2-accurate score estimates, and an O(k/T) bound under intrinsic dimension k with a specific coefficient design.
Problem
Existing convergence guarantees for DDPM samplers in total variation distance are slow (O(sqrt(d/T))) or require strong data/score assumptions (log-concavity, Lipschitz scores, Jacobian accuracy); the goal is a fast rate under minimal assumptions.
Approach
A direct TV-error recursion along the reverse process (instead of Pinsker/KL via Girsanov) using auxiliary truncated chains, with the score-error term entering only through a KL between Gaussian transitions; under a specific coefficient design and a metric-entropy intrinsic dimension, the discretization error scales with k instead of d.
Key results
- Under perfect score estimation, TV(p_X1, p_Y1) <= c d log^3 T / T (ignoring nothing else), improving the prior O(sqrt(d/T)) DDPM rate.
- Score error enters linearly: TV bound adds c * eps_score * sqrt(log T), versus sqrt(d) eps_score for the DDIM analysis of prior work.
- With coefficient design eta_t = 1-alpha_t and sigma_t^2 as in the low-dimensional schedule, the TV bound becomes c k log^3 T / T, improving prior O(sqrt(k/T)) rates.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DDPM with ℓ₂-accurate score estimates under finite first-order moment assumptions instead of the prior DDPM approach | DDPM · General target distributions | TV convergence rate: O(√(d/T)) -> O(d/T) | Helped |
| Use DDPM with coefficients η_t=η_t⋆ and σ_t²=σ_t⋆² | DDPM · Low-dimensional target distributions | TV convergence rate: O(√(k/T)) -> O(k/T) | Helped |
| Use DDPM rather than the compared DDIM sampler | Diffusion sampling with ℓ₂-accurate score estimates · General target distributions, arbitrary T and d | Applicability of the O(d/T) TV convergence rate: O(d/T) when T≫d² -> O(d/T) for arbitrary T and d | Helped |
Surprising findings
- The KL-based route via Pinsker and data processing cannot yield better than O(sqrt(d/T)) for DDPM, since the KL between forward and reverse processes is lower bounded by Omega(d/T) even for a standard Gaussian target, so the authors analyze TV directly.
Limitations
- Assumes perfect or l2-accurate score estimates; the score-matching stage is treated as a black box.
- Requires polynomially bounded first moment M_1 <= T^{c_M}, and the tightness of O(d/T) is left open.
- Bounds are in TV to p_{X_1} rather than the target distribution, and log factors are kept in the bounds.
Experimental setup
Metrics: total variation distance (theoretical bound)
Baselines: DDPM convergence bound O(sqrt(d/T)) (prior work) DDIM convergence bound O(d/T) for T>>d^2 (prior work) low-dimensional DDPM bounds O(sqrt(k/T))
Builds on: DDPM DDIM Pinsker's inequality Girsanov theorem
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