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Denoising diffusions are a powerful method to generate approximate samplesfrom high-dimensional data distributions. Recent results provide polynomialbounds on their convergence rate, assuming $L^2$-accurate scores. Until now,the tightest bounds were either superlinear in the data dimension or requiredstrong smoothness assumptions. We provide the first convergence bounds whichare linear in the data dimension (up to logarithmic factors) assuming onlyfinite second moments of the data distribution. We show that diffusion modelsrequire at most $\tilde O(\frac{d \log^2(1/\delta)}{\varepsilon^2})$ steps toapproximate an arbitrary distribution on $\mathbb{R}^d$ corrupted with Gaussiannoise of variance $\delta$ to within $\varepsilon^2$ in KL divergence. Ourproof extends the Girsanov-based methods of previous works. We introduce arefined treatment of the error from discretizing the reverse SDE inspired bystochastic localization.

Nearly $d$-Linear Convergence Bounds for Diffusion Models via Stochastic Localization