# Score-Optimal Diffusion Schedules

arXiv: https://arxiv.org/abs/2412.07877 · HTML: https://mlsanity.com/paper/2412.07877/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 15 of its 24 references that are in the map.
- First posted on arXiv 10 Dec 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 15 of 15)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Learning to Efficiently Sample from Diffusion Probabilistic Models](https://mlsanity.com/paper/2106.03802/) (2021)
- [Align Your Steps: Optimizing Sampling Schedules in Diffusion Models](https://mlsanity.com/paper/2404.14507/) (2024)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Accelerating Diffusion Sampling with Optimized Time Steps](https://mlsanity.com/paper/2402.17376/) (2024)
- [Blackout Diffusion: Generative Diffusion Models in Discrete-State Spaces](https://mlsanity.com/paper/2305.11089/) (2023)
- [Image generation with shortest path diffusion](https://mlsanity.com/paper/2306.00501/) (2023)

## Built on by (most cited first, 16 of 16)

- [A Fourier Space Perspective on Diffusion Models](https://mlsanity.com/paper/2505.11278/) (2025)
- [Spectral Analysis of Diffusion Models with Application to Schedule Design](https://mlsanity.com/paper/2502.00180/) (2025)
- [The Cosine Schedule is Fisher-Rao-Optimal for Masked Discrete Diffusion Models](https://mlsanity.com/paper/2508.04884/) (2025)
- [Learning Boltzmann Generators via Constrained Mass Transport](https://mlsanity.com/paper/2510.18460/) (2025)
- [Implicit Regularisation in Diffusion Models: An Algorithm-Dependent Generalisation Analysis](https://mlsanity.com/paper/2507.03756/) (2025)
- [Efficient and Unbiased Sampling from Boltzmann Distributions via Variance-Tuned Diffusion Models](https://mlsanity.com/paper/2505.21005/) (2025)
- [Weighted Conditional Flow Matching](https://mlsanity.com/paper/2507.22270/) (2025)
- [Formalizing the Sampling Design Space of Diffusion-Based Generative Models via Adaptive Solvers and Wasserstein-Bounded Timesteps](https://mlsanity.com/paper/2602.12624/) (2026)
- [Metropolis-Adjusted Diffusion Models](https://mlsanity.com/paper/2605.09654/) (2026)
- [Sharpen Your Flow: Sharpness-Aware Sampling for Flow Matching](https://mlsanity.com/paper/2605.11547/) (2026)
- [GAGA: Gaussianity-Aware Gaussian Approximation for Efficient 3D Molecular Generation](https://mlsanity.com/paper/2507.09043/) (2025)
- [Analyzing and Guiding Zero-Shot Posterior Sampling in Diffusion Models](https://mlsanity.com/paper/2602.07715/) (2026)
- [StaTS: Spectral Trajectory Schedule Learning for Adaptive Time Series Forecasting with Frequency Guided Denoiser](https://mlsanity.com/paper/2603.00037/) (2026)
- [From Atoms to Entropy: Optimal Noise Allocation for Diffusion Training in the Convex Regime](https://mlsanity.com/paper/2607.20540/) (2026)
- [A Gaussian Perspective for Distributional Discrepancy in Generative Diffusion Models](https://mlsanity.com/paper/2601.13602/) (2026)
- [The Loss Floor of Denoising Score Matching: Fisher Geometry from Schr"odinger Bridges](https://mlsanity.com/paper/2608.23916/) (2026)

## Abstract (arXiv)

Denoising diffusion models (DDMs) offer a flexible framework for sampling from high dimensional data distributions. DDMs generate a path of probability distributions interpolating between a reference Gaussian distribution and a data distribution by incrementally injecting noise into the data. To numerically simulate the sampling process, a discretisation schedule from the reference back towards clean data must be chosen. An appropriate discretisation schedule is crucial to obtain high quality samples. However, beyond hand crafted heuristics, a general method for choosing this schedule remains elusive. This paper presents a novel algorithm for adaptively selecting an optimal discretisation schedule with respect to a cost that we derive. Our cost measures the work done by the simulation procedure to transport samples from one point in the diffusion path to the next. Our method does not require hyperparameter tuning and adapts to the dynamics and geometry of the diffusion path. Our algorithm only involves the evaluation of the estimated Stein score, making it scalable to existing pre-trained models at inference time and online during training. We find that our learned schedule recovers performant schedules previously only discovered through manual search and obtains competitive FID scores on image datasets.