TaylorGAP: Progressive Anchor Scheduling and Geometry-Aware Taylor Forecasting for Training-Free Diffusion Acceleration

Feature caching accelerates diffusion and flow-matching models by forecasting intermediate features at timesteps that skip a backbone evaluation. TaylorSeer estimates feature derivatives from cached full evaluations, but its fixed refresh interval commits to a full-length forecast block while only the minimum history is available; for a second-order predictor, increasing the offset from one to five enlarges the horizon-dependent factor in the Taylor remainder bound by $5^3=125$. The subsequent transition from unit-spaced startup anchors to the wider refresh interval also creates a non-uniform anchor grid. On such a grid, adjacent lower-order factors have order-dependent effective centers whose separation generally differs from the latest anchor interval, so normalizing the recursive higher-order differences by that interval leaves those factors persistently mis-scaled. Based on these observations, we propose TaylorGAP, a training-free method that couples progressive anchor scheduling with grid-aware Taylor forecasting. Its progressive schedule uses anchor intervals $\{1,2,3,\dots \}$ to delay long extrapolations while preserving a fixed budget of ten full evaluations in a 50-step trajectory. Its geometry-aware recursion normalizes each order by the corresponding effective-center distance; on the progressive grid, this yields the closed-form coefficient ${\lambda }_{r,k}=2k/(2k-r+1)$. The correction is applied only at offsets one and two for stability. At a matched NFE budget, TaylorGAP improves consistently over equal-NFE TaylorSeer for orders two through four on both FLUX.1-dev and HunyuanVideo. Relative to the corresponding baseline, it reduces LPIPS by 28.69%–32.35% on images and 41.00%–43.00% on videos, raises PSNR by 18.28%–19.21% and 26.17%–28.64%, and raises SSIM by 12.23%–13.77% and 13.54%–15.29%. TaylorGAP thus provides a principled, training-free connection between cache scheduling and feature forecasting.