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Unified Convergence Analysis for Score-Based Diffusion Models with Deterministic Samplers
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Develops a unified TV-distance convergence framework for score-based diffusion models with deterministic (ODE-based) samplers, using an ODE analog of Girsanov's theorem, and derives iteration complexity bounds for VP+exponential integrator and VE+DDIM.
Problem
Convergence analysis of score-based diffusion models with deterministic samplers, which cannot use Girsanov's theorem and existing analyses only cover specific forward processes and samplers.
Approach
A lemma bounding the time derivative of TV distance between two ODE flows via drift differences and divergence (an ODE analog of Girsanov), combined with an interpolation trick that turns discrete deterministic samplers (EI, forward Euler, DDIM-type) into continuous ODEs, yielding a five-term error decomposition (score estimation, divergence estimation, score discretization, divergence discretizat
Key results
- For VP forward process with exponential integrator, the sampler reaches TV epsilon in Õ(d^2/ε) iterations under bounded-support and score/divergence error assumptions.
- For VE forward process with DDIM-type sampler, polynomial iteration complexity is established (Õ(d^4/ε^5) per the corollary), the first such result for DDIM with estimated scores.
- Continuous-time reverse ODE bound with OU forward process: TV ≤ O(sqrt(dT + d log(1/δ)) ε_score + ε_div), only logarithmic in δ, versus O(T^{3/4} d^{3/4} δ^{-1} ε^{1/2}) in prior work.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the EI scheme with the VP forward process | deterministic diffusion sampling · VP forward process | iteration complexity: \Tilde{O}({LR^4 d^{2}}/{\epsilon}) | Helped |
| Use the DDIM numerical scheme with the VE forward process | deterministic diffusion sampling · VE forward process | iteration complexity: \Tilde{O}({LR^4 d^{4}}/{\epsilon}^5) | Helped |
| Use the VE forward process rather than the VP forward process | DDIM-type deterministic sampling · VE versus VP forward processes | iteration complexity: \Tilde{O}({LR^4 d^{2}}/{\epsilon}) -> \Tilde{O}({LR^4 d^{4}}/{\epsilon}^5) | Hurt |
| Use a deterministic sampler rather than a stochastic sampler | diffusion sampling · diffusion models | iteration complexity dependence on $\epsilon$: $\epsilon^{-2}$ -> $\epsilon^{-1}$ | Helped |
| Rely on score-estimation error alone, without controlling divergence error | continuous-time reverse ODE · 1-dimensional OU process | total variation distance: \geq \frac{1}{4\pi} | Hurt |
Surprising findings
- Score estimation error alone is insufficient for ODE samplers: a smooth 1-D counterexample has TV at least 1/(4π) between the true and simulated final states regardless of how small the score error is.
Limitations
- VP+EI complexity is quadratic in d, worse than d-linear state-of-the-art ODE bounds.
- VE analysis gives only polynomial bounds due to slow 1/sqrt(T) decay of the prior mismatch.
- Discrete-time analysis requires bounded support of the data distribution.
Experimental setup
Metrics: total variation distance iteration complexity
Baselines: DDIM probability flow ODE analysis (Huang et al. 2024a) stochastic sampler analyses (Girsanov-based)
Builds on: Girsanov-based analysis of diffusion models DDIM probability flow ODE
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