SEEDS: Exponential SDE Solvers for Fast High-Quality Sampling from Diffusion Models

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. We propose Stochastic Explicit 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 use 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 outperform or are competitive with previous SDE solvers. Contrary to the latter, SEEDS are derivative and training free, and we fully prove strong convergence guarantees for them.