Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions
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arXiv
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| Format: | Preprint |
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2024
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| author | Zhang, Kaihong Yin, Caitlyn H. Liang, Feng Liu, Jingbo |
| author_facet | Zhang, Kaihong Yin, Caitlyn H. Liang, Feng Liu, Jingbo |
| contents | We study the asymptotic error of score-based diffusion model sampling in large-sample scenarios from a non-parametric statistics perspective. We show that a kernel-based score estimator achieves an optimal mean square error of $\widetilde{O}\left(n^{-1} t^{-\frac{d+2}{2}}(t^{\frac{d}{2}} \vee 1)\right)$ for the score function of $p_0*\mathcal{N}(0,t\boldsymbol{I}_d)$, where $n$ and $d$ represent the sample size and the dimension, $t$ is bounded above and below by polynomials of $n$, and $p_0$ is an arbitrary sub-Gaussian distribution. As a consequence, this yields an $\widetilde{O}\left(n^{-1/2} t^{-\frac{d}{4}}\right)$ upper bound for the total variation error of the distribution of the sample generated by the diffusion model under a mere sub-Gaussian assumption. If in addition, $p_0$ belongs to the nonparametric family of the $β$-Sobolev space with $β\le 2$, by adopting an early stopping strategy, we obtain that the diffusion model is nearly (up to log factors) minimax optimal. This removes the crucial lower bound assumption on $p_0$ in previous proofs of the minimax optimality of the diffusion model for nonparametric families. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_15602 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions Zhang, Kaihong Yin, Caitlyn H. Liang, Feng Liu, Jingbo Statistics Theory Information Theory Machine Learning We study the asymptotic error of score-based diffusion model sampling in large-sample scenarios from a non-parametric statistics perspective. We show that a kernel-based score estimator achieves an optimal mean square error of $\widetilde{O}\left(n^{-1} t^{-\frac{d+2}{2}}(t^{\frac{d}{2}} \vee 1)\right)$ for the score function of $p_0*\mathcal{N}(0,t\boldsymbol{I}_d)$, where $n$ and $d$ represent the sample size and the dimension, $t$ is bounded above and below by polynomials of $n$, and $p_0$ is an arbitrary sub-Gaussian distribution. As a consequence, this yields an $\widetilde{O}\left(n^{-1/2} t^{-\frac{d}{4}}\right)$ upper bound for the total variation error of the distribution of the sample generated by the diffusion model under a mere sub-Gaussian assumption. If in addition, $p_0$ belongs to the nonparametric family of the $β$-Sobolev space with $β\le 2$, by adopting an early stopping strategy, we obtain that the diffusion model is nearly (up to log factors) minimax optimal. This removes the crucial lower bound assumption on $p_0$ in previous proofs of the minimax optimality of the diffusion model for nonparametric families. |
| title | Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions |
| topic | Statistics Theory Information Theory Machine Learning |
| url | https://arxiv.org/abs/2402.15602 |