Exponential Schedule

The exponential schedule increases the noise variance $\beta_t$ geometrically, following an exponential growth curve. At the beginning, it adds noise a lot slower than the linear schedule—which might sound unintuitive at first, since the name "exponential" suggests rapid growth. But this becomes apparent if you look at the comparative $\beta_t$ in the plot below. During the late steps, the noise curve hits a steep upwards trajectory, rapidly destroying whatever signal is left so it safely reaches pure Gaussian noise at the final step.

By preserving the image longer, the model gets more "useful" timesteps with a healthy mix of signal and noise. In spirit, the exponential and cosine schedule are pursuing this same goal. But they differ in how they achieve it. The exponential schedule is defined directly in terms of $\beta_t$ and the smooth shape of $\bar\alpha_t$ is a consequence of its geometric progression. In contrast, the cosine schedule defines $\bar\alpha_t$ directly to have the desired shape.