Learning the reverse-process variance as an interpolation between two fixed endpoints improves likelihood

The original DDPM fixes the reverse variance to a constant (either beta_t or the lower-bound beta_tilde_t), finding that learning a full diagonal variance leads to unstable training and poorer samples. Nichol and Dhariwal (2021) proposed a compromise: predict a mixing vector v that interpolates between the two variance bounds in log space. Since L_simple does not depend on the variance, they use a hybrid objective L_hybrid = L_simple + lambda*L_VLB (with lambda=0.001), stopping gradient through the mean so that L_VLB guides only the variance. This achieves better negative log-likelihood while keeping the sample quality of L_simple. Because naive L_VLB gradients are too noisy for stable variance learning, they use an importance-sampled, time-averaged smoothed version of L_VLB.

Examples

With lambda=0.001 the variance term is a small correction; the gradient is stopped through the mean prediction so L_VLB only trains the variance output.

Assessment

Why can L_simple alone not train a good variance prediction? Describe the hybrid-objective strategy and explain what improvement it achieves.