Smoothed-score queries to the resolvent of a Gaussian precision matrix yield an O((log κ) log(1/δ)) query sampler for TV error δ and an Ω(log κ) bit lower bound, improving the condition-number dependence from √κ.
High-accuracy sampling for diffusion models and log-concave distributions
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abstract
We present algorithms for diffusion model sampling which obtain $\delta$-error in $\mathrm{polylog}(1/\delta)$ steps, given access to $\widetilde O(\delta)$-accurate score estimates in $L^2$. This is an exponential improvement over all previous results. Specifically, under minimal data assumptions, the complexity is $\widetilde O(d_\star \mathrm{polylog}(1/\delta))$ where $d_\star$ is the intrinsic dimension of the data. Further, under a non-uniform $L$-Lipschitz condition, the complexity reduces to $\widetilde O(L \mathrm{polylog}(1/\delta))$. Our approach also yields the first $\mathrm{polylog}(1/\delta)$ complexity sampler for general log-concave distributions using only gradient evaluations.
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Diffusion sampling from d-dimensional distributions requires at least ~sqrt(d) adaptive score queries when score estimates have polynomial accuracy.
Metropolis-adjusted Langevin correctors using score-based acceptance probabilities, including an exact Bernoulli factory method and a Simpson's rule approximation, reduce sampling bias in diffusion models and improve FID scores.
A proximal gradient sampler for composite log-concave distributions achieves near-optimal iteration complexity of order kappa sqrt(d) log^4(1/epsilon) in total variation distance under strong convexity and smoothness.
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Smoothed Score Queries and the Complexity of Sampling
Smoothed-score queries to the resolvent of a Gaussian precision matrix yield an O((log κ) log(1/δ)) query sampler for TV error δ and an Ω(log κ) bit lower bound, improving the condition-number dependence from √κ.
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Query Lower Bounds for Diffusion Sampling
Diffusion sampling from d-dimensional distributions requires at least ~sqrt(d) adaptive score queries when score estimates have polynomial accuracy.
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Metropolis-Adjusted Diffusion Models
Metropolis-adjusted Langevin correctors using score-based acceptance probabilities, including an exact Bernoulli factory method and a Simpson's rule approximation, reduce sampling bias in diffusion models and improve FID scores.
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A proximal gradient algorithm for composite log-concave sampling
A proximal gradient sampler for composite log-concave distributions achieves near-optimal iteration complexity of order kappa sqrt(d) log^4(1/epsilon) in total variation distance under strong convexity and smoothness.