Convergence for score-based generative modeling with polynomial complexity
This addresses the theoretical foundation for generative modeling, offering rigorous guarantees that could improve reliability and efficiency in applications like image synthesis, though it is incremental in providing proofs rather than new methods.
The paper proves the first polynomial convergence guarantees for score-based generative modeling, showing that samples can be drawn from a smooth distribution with error depending polynomially on its log-Sobolev constant, without exponential time or dimensionality issues. It also provides theoretical grounding for annealing and predictor-corrector algorithms in practice.
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.