Improved Discretization Complexity Analysis of Consistency Models: Variance Exploding Forward Process and Decay Discretization Scheme

Consistency models, a new class of one-step generative models, have shown competitive performance with multi-step diffusion models. The most challenging part of consistency models is the training process, which discretizes the continuous diffusion process into K steps and trains a one-step mapping function on these discretized timepoints. Despite the empirical success, only a few works focus on the discretization complexity K, and their setting is far from that of empirical works. More specifically, the current theoretical works analyze the variance preserving (VP) diffusion process with a uniform stepsize, while empirical works adopt a variance exploding (VE) process with a decay discretization stepsize. As a result, these works suffer from large discretization complexity and fail to explain the empirical success of consistency models. To close the gap between theory and application, we analyze consistency models with (1) VE process and (2) decay stepsize and prove the state-of-the-art discretization complexity for consistency models. This result is competitive with the results of diffusion models and shows the potential of consistency models. To balance the computation and performance, previous empirical work further proposes a 2-step consistency algorithm. In this work, we also analyze the role of 2-step sampling and show that it improves the discretization complexity compared with one-step generation.