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The analysis supports annealing for warm starts and proves predictor-corrector schemes converge better than using either part alone.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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prove about score-based generative modeling?","Question",{"text":63,"@type":64},"It provides the first polynomial convergence guarantees for the core score-based sampling mechanism when the score estimate is accurate in L2(p).","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How does the convergence guarantee avoid common failure modes?",{"text":68,"@type":64},"It avoids errors that grow exponentially in time and avoids the curse of dimensionality compared with prior results.",{"name":70,"@type":61,"acceptedAnswer":71},"Why does the paper emphasize annealing and predictor-corrector methods?",{"text":72,"@type":64},"The proof relies on annealing to obtain a warm start at each step, and the analysis shows predictor-corrector algorithms achieve better convergence than using either component 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\nDepartment of Applied Mathematics and Statistics Johns Hopkins University [hlee283@jhu.edu](hlee283@jhu.edu)  \nJianfeng Lu  \nDepartment of Mathematics Duke University [jianfeng@math.duke.edu](jianfeng@math.duke.edu)  \nYixin Tan  \nDepartment of Mathematics  \nDuke University  \n[yixin.tan@duke.edu](yixin.tan@duke.edu)  \nAbstract  \nScore-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:  \ndrawing samples from a probability density p given a score estimate (an estimate of ∇lnp) that is accurate in L2 (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 polynomiallyon 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.  \n1 Introduction  \nA key task in machine learning is to learn a probability distribution from data, in a way that allows efficient generation of additional samples from the learned distribution. Score-based generative modeling (SGM) is one empirically successful approach that implicitly learns the probability distribution by learning how to transform white noise into the data distribution, and gives stateof-the-art performance for generating images and audio [SE19; Dat+19; Gra+19; SE20; Son+20b; Men+21; Son+21b; Son+21a; Jin+22] . It also yields a conditional generation process for inverse problems [DN21] . The basic idea behind score-based generative modeling is to first estimate the score function from data [Son+20a] and then to sample the distribution based on the learned score function. Other approaches for generative modeling include generative adversarial networks (GANs)[Goo+14; ACB17], normalizing flows [DSB16], variational autoencoders [KW19], and energybased models [ZML16] . While score-based generative modeling has achieved great success, its theoretical analysis is still lacking and is the focus of our work.  \n36th Conference on Neural Information Processing Systems (NeurIPS 2022) .  \n1.1 Background  \nGeneral framework. The score function of a distribution P with density p is defined as the gradient of the log-pdf, ∇lnp. Its significance arises from the fact that knowing the score function allows running a variety of sampling algorithms, based on discretizations of stochastic differential equations (SDE’s), to sample from p. SGM consists of two steps: first, learning an estimate of the score function for a sequence of “noisy” versions of the data distribution Pdata , and second, using the score function in lieu of the gradient of the log-pdf in the chosen sampling algorithm. We now describe each of these steps more precisely.  \nFirst, a method of adding noise to the data distribution is fixed; this takes the form of evolving a (forward) stochastic differential equation (SDE) starting from the data distribution. We fix a sequence of noise levels σ 1 \u003C ··· \u003C σN . For σ ∈ {σ1 ,...,σN }, let the resulting distributions be Pσ2 and the distributions conditional on the starting data point be Pσ2 (·|x) . Typically, σ 1 is chosen so that Pσ21 ≈ Pdata and Pσ2N is close to some “prior” distribution that is easy to sample from, such as N(0,σ2NId) . While the score ∇lnpσ2 cannot be estimate","cbCaitfiMdF4gCr7","https://ap.wps.com/l/cbCaitfiMdF4gCr7","pdf",328438,13,"English","# Introduction\n## Background\n### General framework\n## Examples of SGM","[{\"question\":\"What does the paper prove about score-based generative modeling?\",\"answer\":\"It provides the first polynomial convergence guarantees for the core score-based sampling mechanism when the score estimate is accurate in L2(p).\"},{\"question\":\"How does the convergence guarantee avoid common failure modes?\",\"answer\":\"It avoids errors that grow exponentially in time and avoids the curse of dimensionality compared with prior results.\"},{\"question\":\"Why does the paper emphasize annealing and predictor-corrector methods?\",\"answer\":\"The proof relies on annealing to obtain a warm start at each step, and the analysis shows predictor-corrector algorithms achieve better convergence than using either component alone.\"}]","Convergence for score-based generative modeling with polynomial complexity | PDF",33]