Convergence for score-based generative modeling with polynomial complexity

• URL: http://arxiv.org/abs/2206.06227v2
• Date: Wed, 3 May 2023 17:51:05 GMT
• Title: Convergence for score-based generative modeling with polynomial complexity
• Authors: Holden Lee and Jianfeng Lu and Yixin Tan
• Abstract summary: We prove the first convergence guarantees for the core mechanic behind Score-based generative modeling. Compared to previous works, we do not incur error that grows exponentially in time or that suffers from a curse of dimensionality. We show that a predictor-corrector gives better convergence than using either portion alone.
• Score: 9.953088581242845
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Score-based generative modeling (SGM) is a highly successful approach for learning a probability distribution from data and generating further samples. We prove the first polynomial convergence guarantees for the core mechanic behind SGM: drawing samples from a probability density $p$ given a score estimate (an estimate of $\nabla \ln p$) that is accurate in $L^2(p)$. Compared to previous works, we do not incur error that grows exponentially in time or that suffers from a curse of dimensionality. Our guarantee works for any smooth distribution and depends polynomially on its log-Sobolev constant. Using our guarantee, we give a theoretical analysis of score-based generative modeling, which transforms white-noise input into samples from a learned data distribution given score estimates at different noise scales. Our analysis gives theoretical grounding to the observation that an annealed procedure is required in practice to generate good samples, as our proof depends essentially on using annealing to obtain a warm start at each step. Moreover, we show that a predictor-corrector algorithm gives better convergence than using either portion alone.

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