Noise Scheduling as Information-Guided Allocation in Diffusion Training

Gabriel Raya; Bac Nguyen; Georgios Batzolis; Yuhta Takida; Dejan Stancevic; Naoki Murata; Chieh-Hsin Lai; Yuki Mitsufuji

cs.LGFeb 20, 2026

Noise Scheduling as Information-Guided Allocation in Diffusion Training

Authors: Gabriel Raya, Bac Nguyen, Georgios Batzolis, Yuhta Takida, Dejan Stancevic, Naoki Murata, Chieh-Hsin Lai, Yuki Mitsufuji,

Organizations: Tilburg University & JADS · Sony AI · University of Cambridge · Radboud University ·

Abstract

We introduce InfoNoise, an online adaptive noise schedule for diffusion training that reallocates optimization effort toward noise levels where denoising is most informative. Together with loss weighting, a noise schedule induces an effective allocation across denoising problems, often fixed before informative noise levels are known. InfoNoise makes this allocation data-adaptive by estimating a conditional-entropy-rate profile from denoising losses during training, without auxiliary models or offline search. Through I--MMSE, this profile identifies where noisy observations rapidly reduce uncertainty about the clean sample and guides adaptation of the training noise distribution. It changes only this distribution, keeping the objective, weighting, and parameterization fixed. On image benchmarks, where schedules have been extensively tuned, InfoNoise matches or slightly exceeds strong baselines and can reach the same quality with fewer updates. On representation, sequence, and modality shifts, including DNA and language generation, InfoNoise improves over fixed and adaptive baselines and reaches target quality with up to $3\times$ less training compute. These results establish the conditional-entropy-rate profile as the data-dependent target for noise schedule design and make online adaptation a practical alternative to manual schedule search.

Figures & tables

Figure 1: The information profile defines the allocation target in diffusion training. (a) In an analytical two-state Gaussian channel, uncertainty is not resolved uniformly along the path. The entropy-rate profile peaks at the decision window where noisy observations most rapidly reveal the clean state; the derivation is given in Section B.5 . (b) In mature image regimes, inherited allocations already overlap the measured informative region, so InfoNoise preserves the allocation while recovering this region online. (c) Under shift, the information profile moves while inherited allocation remains fixed, producing allocation mismatch. InfoNoise estimates the shifted profile online and redirects allocation toward the region where uncertainty is resolved, without noise-schedule retuning.

Figure 2: Entropy rate localizes uncertainty resolution. MMSE measures residual posterior uncertainty; entropy rate identifies where that uncertainty changes along the path.

Figure 3: Inverse-CDF sampling from the adapted density. Shown on CIFAR-10 for clarity, the adapted density

\pi

defines a cumulative mass

F_{\pi}

. Uniform samples are mapped through

F_{\pi}^{-1}

to draw training noise levels. The same construction applies unchanged in other domains.

Figure 4: InfoNoise converts an online information estimate into a training noise distribution. (A) Unweighted losses track

x

-space denoising error, which the path factor converts into an online information-profile estimate. (B) With fixed loss weights, InfoNoise samples from

\pi\propto\hat{\rho}/w

, not directly from

\hat{\rho}

. (C) The induced allocation

\pi w

matches the profile while the objective and weighting remain unchanged. Noise levels are drawn from

\pi

by inverse-CDF sampling, illustrated in Figure 3 .

Figure 5: InfoNoise reaches the best baseline quality with fewer training updates. For each dataset, the target is the final matched-budget quality of the strongest non- InfoNoise baseline, including fixed schedules and KG-adaptive. Speedup reports how much less training compute InfoNoise needs to reach that target. A value of

1.00

means no measured reduction within the training budget; values above

1.00

indicate that the same quality is reached with fewer updates.

Dataset	Metric	NFE	Log-unif.	Cosine	FM-OT	EDM	KG-adapt.	InfoNoise	$\times$ vs fixed
			Fixed schedules				Adaptive schedules		Efficiency
MNIST	FID $\downarrow$	35	$1.02$	$0.46$	$0.44$	$0.44$	0.39	0.41	1.20
FashionMNIST	FID $\downarrow$	35	$2.75$	$1.76$	$1.93$	$1.78$	$1.82$	1.71	1.45
CIFAR-10 uncond.	FID $\downarrow$	35	$5.93$	$2.16$	$2.31$	$2.04$	$2.00$	1.98	1.40
CIFAR-10 cond.	FID $\downarrow$	35	$4.45$	$2.11$	$2.13$	$1.85$	1.84	1.84	1.50
FFHQ $64{\times}64$	FID $\downarrow$	79	$4.68$	$2.69$	$2.72$	2.53	$2.54$	2.53	1.00

Table 1: Information-guided allocation automates schedule adaptation across regimes. Each row compares schedules trained to the same budget and evaluated at the listed NFE. Quality is the final metric at that budget. Efficiency reports the compute InfoNoise needs to reach the final metric of the best competing baseline; values above one mean the same target is reached with fewer updates. Shading denotes InfoNoise ; bold marks best final quality and speedups above one.

Figure 6: Profile alignment explains schedule transfer. Post hoc information profiles are compared with the effective allocation induced by inherited EDM sampling and by InfoNoise . (A) In matched image regimes, inherited allocation overlaps the informative region, and InfoNoise converges to a similar allocation online. (B) Under shift, the information profile moves away from inherited image allocation. InfoNoise moves the induced allocation toward the shifted profile without schedule retuning. Profiles are used only for analysis and never provided to InfoNoise during training.

Figure 7: Schedule search is allocation search. (a) Sweeping

P_{\mathrm{mean}}

in the EDM log-normal schedule shifts effective allocation along the DNA corruption path while other training choices remain fixed. InfoNoise reaches the informative region online in one run. (b) Allocations closer to the information profile reach the strongest non- InfoNoise baseline target with fewer updates. (c) Training compute decreases as effective allocation becomes better aligned with the DNA information profile, showing that misalignment with the profile is what makes schedule transfer expensive.

Warm-up prior	Sei-FID $\downarrow$
EDM log-normal	0.460
Log-uniform in $\sigma$	0.463
Log- $\sigma$ const.-entropy	0.456
Spread	$0.007$

Table 2: Warm-up does not set final performance. DNA Sei-FID changes by only

0.007

across warm-up priors.

Figure 8: Training-allocation gains persist under fixed CFG. With guidance scale and sampler fixed, InfoNoise reaches higher MAUVE earlier.

Figure 9: InfoNoise rapidly recovers the information profile and stabilizes. On DNA, the online estimate moves from the warm-up allocation toward the post hoc diagnostic profile within the first rebuilds. The diagnostic profile is computed after training and never provided to InfoNoise . Sampler-CDF changes fall below

10^{-2}

, showing that the sampler stabilizes quickly rather than tracking transient loss fluctuations.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 10: Baseline schedules induce different effective allocations. Curves are expressed in the common

x

-prediction view on the shared log-SNR support. The plotted allocation combines the sampling density and the

x

-space loss weight,

\phi_{x}(\lambda)=\pi(\lambda)w_{x}(\lambda)

Baseline	Sampling rule	$\pi(\lambda)$	$\phi_{x}(\lambda)$
EDM	$\log\sigma\sim\mathcal{N}(P_{\rm mean},P_{\rm std}^{2})$	$\mathcal{N}(\lambda;-2P_{\rm mean},(2P_{\rm std})^{2})$	$\propto\pi_{\rm EDM}(\lambda)(1+\sigma_{\rm data}^{2}e^{\lambda})$
Log-uniform	$\log\sigma\sim\mathcal{U}$	const.	$\propto w_{x}(\lambda)$
Cosine / iDDPM	$t\sim\mathcal{U}(0,1)$	$\propto\sech(\lambda/2)$	$\propto\pi_{\rm cos}(\lambda)w_{x}(\lambda)$
FM-OT	$t\sim\mathcal{U}(0,1)$	$\propto\sech^{2}(\lambda/4)$	$\propto e^{\lambda/2}$

Appendix

Table 3: Baseline allocations in the common

x

-prediction view. Densities are with respect to

\lambda=\log\mathrm{SNR}

and are shown up to normalization on the common support. Symbolic

w_{x}(\lambda)

denotes the fixed

x

-space loss weight of the corresponding experiment.

Figure 11: Entropy rate localizes the decision window. MMSE measures remaining uncertainty; entropy rate marks where that uncertainty changes fastest. Fixed-point branches show the same transition geometrically.

Figure 12: The data distribution sets the information window. Under the same VE path, changing the atom scale $a$ in $x_{0}\in\{-a,+a\}$ shifts where posterior uncertainty is resolved. (a,b) Likelihood geometry for $a=1$ and $a=6$ . Bayes-denoiser fixed points move along the noise axis as data scale changes. (c) The entropy-rate density shifts with $a$ , showing that the informative region is induced by the data distribution under the corruption path, not the noise coordinate alone.

Figure 13: Low-noise boundary structure motivates gated entropy-rate profiles. Continuous domains show low-noise entropy-rate growth, while discrete domains show extended low-noise power-law structure in the raw estimate. Vertical markers indicate the detected pivot

c

, the upper edge of the boundary-dominated region. The gate suppresses endpoint structure before normalization while preserving the interior information-bearing region.

Asset	Use in this paper	License / terms
CIFAR-10 ( Krizhevsky and Hinton, 2009 )	Image and bitstream experiments	Public benchmark dataset; original Toronto page describes the dataset but does not state an explicit license
FFHQ ( Karras et al., 2019 )	FFHQ $64{\times}64$ image experiments	Images retain their original Flickr licenses; dataset metadata and tooling are released under the terms stated by the FFHQ source
OpenWebText-2 ( Gao et al., 2020 )	Text bitstream experiments	MIT license, as stated by the EleutherAI OpenWebText2 repository
EDM code and setup ( Karras et al., 2022 )	Image baseline, preconditioning, loss weighting, and sampler	Official NVIDIA research code release; repository includes a license file and directs licensing inquiries to NVIDIA Research Licensing
MAUVE ( Pillutla et al., 2021 )	Text evaluation metric	GPLv3, as stated by the mauve-text package metadata
GPT-2-large ( Radford et al., 2019 )	Feature extractor for MAUVE	Modified MIT License, as stated by the model release

Appendix

Table 4: Existing assets used in the experiments. License and terms entries follow the corresponding source documentation where available.

Figure 14: InfoNoise turns online loss statistics into effective allocation. (A) Unweighted denoising losses measure the model’s

x

-space error across noise levels. (B) The path factor converts these losses into an online information-profile estimate, shown alongside a fixed reference sampler and a post hoc diagnostic profile. (C) Rebuilding the sampler changes the induced effective allocation while the objective and loss weighting remain fixed.

Panel of Figure 15 — Figure 15: InfoNoise estimates dataset-specific information profiles. Each panel shows the normalized online profile estimate and the inverse-CDF sampler induced by InfoNoise . The sampler $\pi(\sigma)$ , together with the fixed per-noise weight $w(\sigma)$ , determines the realized allocation $\phi(\sigma)=\pi(\sigma)w(\sigma)$ . The profiles show that the informative region shifts across image, discrete, and sequence regimes.

Panel of Figure 16 — Figure 16: Qualitative samples at matched training checkpoints. InfoNoise and EDM are evaluated with the same solver, inference grid, and NFE. Checkpoints are matched by processed examples.

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Noise Schedule Design for Diffusion Models: An Optimal Control Perspective

Seo Taek Kong, Weina Wang, R. Srikant

ECE & CSL, University of Illinois Urbana-Champaign · Computer Science Department, Carnegie Mellon University · ECE, CSL & NCSA, University of Illinois Urbana-Champaign

Diffusion Sampling Diffusion-Based Image Generation Diffusion Timestep Scheduling Optimal Control Diffusion Models

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We develop a principled framework for analyzing and designing noise schedules in diffusion models. We show that one can recast this design problem as an optimal control problem, whose state is the Fisher information of the diffusion process, which evolves according to an ODE, and whose control input is the noise schedule. The objective of the optimal control problem is a functional involving the Fisher information, which is shown to be an upper bound on the Kullback-Leibler sampling error. By solving this optimal control problem, we obtain sufficient conditions on noise schedules under which

\tilde{\mathcal{O}}(d/n)

sampling error is achievable, where

d

is the data dimension and

n

is the number of discretization steps. While existing theoretical works also prove that

\tilde{\mathcal{O}}(d/n)

sampling error bounds are achievable, these results hold for specific noise schedules that differ from those used in practice. Under additional assumptions on the data distribution, we derive a family of noise schedules that recovers the functional forms of schedules widely used in practice. We show that these schedules achieve the

\tilde{\mathcal{O}}(d/n)

error bound when their parameters are chosen according to our analysis. Our theory offers a principled approach to noise schedule design, yielding explicit functional forms and guidance for selecting their parameters.

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