A potent class of generative models known as Diffusion Probabilistic Models (DPMs) has become prominent. A forward diffusion process adds gradually noise to data, while a model learns to gradually denoise. Sampling from pre-trained DPMs is obtained by solving differential equations (DE) defined by the learnt model, a process which has shown to be prohibitively slow. Numerous efforts on speeding-up this process have consisted on crafting powerful ODE solvers. Despite being quick, such solvers do not usually reach the optimal quality achieved by available slow SDE solvers. Our goal is to propose SDE solvers that reach optimal quality without requiring several hundreds or thousands of NFEs to achieve that goal. In this work, we propose Stochastic Exponential Derivative-free Solvers (SEEDS), improving and generalizing Exponential Integrator approaches to the stochastic case on several frameworks. After carefully analyzing the formulation of exact solutions of diffusion SDEs, we craft SEEDS to analytically compute the linear part of such solutions. Inspired by the Exponential Time-Differencing method, SEEDS uses a novel treatment of the stochastic components of solutions, enabling the analytical computation of their variance, and contains high-order terms allowing to reach optimal quality sampling $\sim3$-$5\times$ faster than previous SDE methods. We validate our approach on several image generation benchmarks, showing that SEEDS outperforms or is competitive with previous SDE solvers. Contrary to the latter, SEEDS are derivative and training free, and we fully prove strong convergence guarantees for them.

翻译：一类被称为扩散概率模型（DPMs）的强效生成模型已日益凸显。前向扩散过程逐步向数据添加噪声，而模型则学习逐步去噪。从预训练DPMs中采样需通过求解由学习模型定义的微分方程（DE）实现，该过程已被证明极其缓慢。为加速此过程，大量研究致力于构建高效的ODE求解器。尽管速度较快，这类求解器通常无法达到现有慢速SDE求解器所实现的最优质量。我们的目标是提出能在无需数百或数千次函数评估（NFEs）条件下达到最优质量的SDE求解器。本文提出随机指数型免导数求解器（SEEDS），在多个框架下改进并推广指数积分器方法至随机情形。通过深入分析扩散SDE精确解的公式化表达，我们设计SEEDS以解析计算此类解的线性部分。受指数时间差分法启发，SEEDS采用新型处理方法应对解的随机分量，实现其方差的解析计算，并包含高阶项以达到最优质量采样，速度较先前SDE方法提升约3-5倍。我们在多个图像生成基准上验证了该方法，表明SEEDS优于或媲美现有SDE求解器。与后者不同，SEEDS免导数且无需训练，我们为其提供了完整的强收敛性保证。