Common Diffusion Noise Schedules and Sample Steps are Flawed
akhaliq
Abstract
We discover that common diffusion noise schedules do not enforce the last timestep to have zero signal-to-noise ratio (SNR), and some implementations of diffusion samplers do not start from the last timestep. Such designs are flawed and do not reflect the fact that the model is given pure Gaussian noise at inference, creating a discrepancy between training and inference. We show that the flawed design causes real problems in existing implementations. In Stable Diffusion, it severely limits the model to only generate images with medium brightness and prevents it from generating very bright and dark samples. We propose a few simple fixes: (1) rescale the noise schedule to enforce zero terminal SNR; (2) train the model with v prediction; (3) change the sampler to always start from the last timestep; (4) rescale classifier-free guidance to prevent over-exposure. These simple changes ensure the diffusion process is congruent between training and inference and allow the model to generate samples more faithful to the original data distribution.
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Schedule to Zero Terminal SNR is not working, because it gets inf at last timestep.
sqrt_recip_alphas_cumprod tensor([ 1.0003, 1.0009, 1.0017, 1.0027, 1.0040, 1.0056, 1.0074, 1.0094,
1.0117, 1.0143, 1.0171, 1.0201, 1.0235, 1.0271, 1.0310, 1.0352,
1.0397, 1.0444, 1.0495, 1.0549, 1.0605, 1.0665, 1.0729, 1.0795,
1.0866, 1.0939, 1.1017, 1.1098, 1.1184, 1.1273, 1.1367, 1.1464,
1.1567, 1.1674, 1.1786, 1.1903, 1.2026, 1.2154, 1.2288, 1.2428,
1.2574, 1.2726, 1.2886, 1.3053, 1.3228, 1.3410, 1.3601, 1.3801,
1.4011, 1.4230, 1.4460, 1.4701, 1.4954, 1.5220, 1.5499, 1.5792,
1.6101, 1.6426, 1.6768, 1.7129, 1.7511, 1.7915, 1.8342, 1.8794,
1.9275, 1.9785, 2.0329, 2.0908, 2.1526, 2.2188, 2.2898, 2.3660,
2.4481, 2.5368, 2.6327, 2.7370, 2.8505, 2.9746, 3.1108, 3.2608,
3.4270, 3.6120, 3.8190, 4.0522, 4.3170, 4.6199, 4.9698, 5.3785,
5.8620, 6.4427, 7.1530, 8.0416, 9.1848, 10.7100, 12.8463, 16.0520,
21.3966, 32.0883, 64.1689, inf], dtype=torch.float64)
sqrt_recipm1_alphas_cumprod tensor([2.5133e-02, 4.1840e-02, 5.7956e-02, 7.3890e-02, 8.9761e-02, 1.0562e-01,
1.2150e-01, 1.3742e-01, 1.5340e-01, 1.6944e-01, 1.8555e-01, 2.0175e-01,
2.1805e-01, 2.3446e-01, 2.5099e-01, 2.6764e-01, 2.8443e-01, 3.0136e-01,
3.1846e-01, 3.3572e-01, 3.5317e-01, 3.7081e-01, 3.8865e-01, 4.0671e-01,
4.2499e-01, 4.4353e-01, 4.6231e-01, 4.8138e-01, 5.0072e-01, 5.2038e-01,
5.4035e-01, 5.6066e-01, 5.8133e-01, 6.0238e-01, 6.2383e-01, 6.4570e-01,
6.6801e-01, 6.9079e-01, 7.1407e-01, 7.3787e-01, 7.6223e-01, 7.8717e-01,
8.1273e-01, 8.3894e-01, 8.6585e-01, 8.9350e-01, 9.2193e-01, 9.5118e-01,
9.8132e-01, 1.0124e+00, 1.0445e+00, 1.0776e+00, 1.1118e+00, 1.1473e+00,
1.1841e+00, 1.2222e+00, 1.2618e+00, 1.3031e+00, 1.3460e+00, 1.3908e+00,
1.4375e+00, 1.4864e+00, 1.5376e+00, 1.5913e+00, 1.6478e+00, 1.7072e+00,
1.7699e+00, 1.8361e+00, 1.9063e+00, 1.9807e+00, 2.0599e+00, 2.1443e+00,
2.2346e+00, 2.3314e+00, 2.4354e+00, 2.5477e+00, 2.6693e+00, 2.8014e+00,
2.9456e+00, 3.1037e+00, 3.2779e+00, 3.4708e+00, 3.6858e+00, 3.9269e+00,
4.1995e+00, 4.5103e+00, 4.8681e+00, 5.2847e+00, 5.7760e+00, 6.3646e+00,
7.0828e+00, 7.9792e+00, 9.1302e+00, 1.0663e+01, 1.2807e+01, 1.6021e+01,
2.1373e+01, 3.2073e+01, 6.4161e+01, inf], dtype=torch.float64)
Schedule to Zero Terminal SNR is not working, because it gets inf at last timestep.
sqrt_recip_alphas_cumprod tensor([ 1.0003, 1.0009, 1.0017, 1.0027, 1.0040, 1.0056, 1.0074, 1.0094,
1.0117, 1.0143, 1.0171, 1.0201, 1.0235, 1.0271, 1.0310, 1.0352,
1.0397, 1.0444, 1.0495, 1.0549, 1.0605, 1.0665, 1.0729, 1.0795,
1.0866, 1.0939, 1.1017, 1.1098, 1.1184, 1.1273, 1.1367, 1.1464,
1.1567, 1.1674, 1.1786, 1.1903, 1.2026, 1.2154, 1.2288, 1.2428,
1.2574, 1.2726, 1.2886, 1.3053, 1.3228, 1.3410, 1.3601, 1.3801,
1.4011, 1.4230, 1.4460, 1.4701, 1.4954, 1.5220, 1.5499, 1.5792,
1.6101, 1.6426, 1.6768, 1.7129, 1.7511, 1.7915, 1.8342, 1.8794,
1.9275, 1.9785, 2.0329, 2.0908, 2.1526, 2.2188, 2.2898, 2.3660,
2.4481, 2.5368, 2.6327, 2.7370, 2.8505, 2.9746, 3.1108, 3.2608,
3.4270, 3.6120, 3.8190, 4.0522, 4.3170, 4.6199, 4.9698, 5.3785,
5.8620, 6.4427, 7.1530, 8.0416, 9.1848, 10.7100, 12.8463, 16.0520,
21.3966, 32.0883, 64.1689, inf], dtype=torch.float64)
sqrt_recipm1_alphas_cumprod tensor([2.5133e-02, 4.1840e-02, 5.7956e-02, 7.3890e-02, 8.9761e-02, 1.0562e-01,
1.2150e-01, 1.3742e-01, 1.5340e-01, 1.6944e-01, 1.8555e-01, 2.0175e-01,
2.1805e-01, 2.3446e-01, 2.5099e-01, 2.6764e-01, 2.8443e-01, 3.0136e-01,
3.1846e-01, 3.3572e-01, 3.5317e-01, 3.7081e-01, 3.8865e-01, 4.0671e-01,
4.2499e-01, 4.4353e-01, 4.6231e-01, 4.8138e-01, 5.0072e-01, 5.2038e-01,
5.4035e-01, 5.6066e-01, 5.8133e-01, 6.0238e-01, 6.2383e-01, 6.4570e-01,
6.6801e-01, 6.9079e-01, 7.1407e-01, 7.3787e-01, 7.6223e-01, 7.8717e-01,
8.1273e-01, 8.3894e-01, 8.6585e-01, 8.9350e-01, 9.2193e-01, 9.5118e-01,
9.8132e-01, 1.0124e+00, 1.0445e+00, 1.0776e+00, 1.1118e+00, 1.1473e+00,
1.1841e+00, 1.2222e+00, 1.2618e+00, 1.3031e+00, 1.3460e+00, 1.3908e+00,
1.4375e+00, 1.4864e+00, 1.5376e+00, 1.5913e+00, 1.6478e+00, 1.7072e+00,
1.7699e+00, 1.8361e+00, 1.9063e+00, 1.9807e+00, 2.0599e+00, 2.1443e+00,
2.2346e+00, 2.3314e+00, 2.4354e+00, 2.5477e+00, 2.6693e+00, 2.8014e+00,
2.9456e+00, 3.1037e+00, 3.2779e+00, 3.4708e+00, 3.6858e+00, 3.9269e+00,
4.1995e+00, 4.5103e+00, 4.8681e+00, 5.2847e+00, 5.7760e+00, 6.3646e+00,
7.0828e+00, 7.9792e+00, 9.1302e+00, 1.0663e+01, 1.2807e+01, 1.6021e+01,
2.1373e+01, 3.2073e+01, 6.4161e+01, inf], dtype=torch.float64)
Zero terminal SNR itself does not have problems and can totally work. The problem is that a lot of the samplers' math formulations are derived assuming model epsilon prediction, so many implementation will convert model output to epsilon first, then perform the math calculation. But epsilon prediction will never work with Zero Terminal SNR, hence the undefined division error. The solution is to directly derive math from v prediction or x0 prediction.
DDPM, DDIM implementation in diffusers should totally work. What sampler are you using?
Yes I understand from the paper it's only working with v by default and with noise is nonsense. I test it in my own environment (Jupyter notebook). I'm using DDIM. The authors was talking about v prediction but not impossibilities with another configuration. Thanks.
Can the ZeroSNR method work correctly in EulerDiscreteScheduler or EulerAncestralDiscreteScheduler ?
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