# Steering diffusion models with quadratic rewards: a fine-grained analysis

arXiv: https://arxiv.org/abs/2602.16570 · HTML: https://mlsanity.com/paper/2602.16570/

## Key facts

- Cited by 5 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 27 references that are in the map.
- First posted on arXiv 18 Feb 2026.

## Placed in

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

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

- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [A General Framework for Inference-time Scaling and Steering of Diffusion Models](https://mlsanity.com/paper/2501.06848/) (2025)
- [From Decoding to Meta-Generation: Inference-time Algorithms for Large Language Models](https://mlsanity.com/paper/2406.16838/) (2024)
- [RL with KL penalties is better viewed as Bayesian inference](https://mlsanity.com/paper/2205.11275/) (2022)
- [What does guidance do? A fine-grained analysis in a simple setting](https://mlsanity.com/paper/2409.13074/) (2024)
- [Diffusion Posterior Sampling is Computationally Intractable](https://mlsanity.com/paper/2402.12727/) (2024)
- [Representation-Based Exploration for Language Models: From Test-Time to Post-Training](https://mlsanity.com/paper/2510.11686/) (2025)
- [Taming Imperfect Process Verifiers: A Sampling Perspective on Backtracking](https://mlsanity.com/paper/2510.03149/) (2025)
- [Guided Speculative Inference for Efficient Test-Time Alignment of LLMs](https://mlsanity.com/paper/2506.04118/) (2025)
- [Sampling Approximately Low-Rank Ising Models: MCMC meets Variational Methods](https://mlsanity.com/paper/2202.08907/) (2022)
- [ReGuidance: A Simple Diffusion Wrapper for Boosting Sample Quality on Hard Inverse Problems](https://mlsanity.com/paper/2506.10955/) (2025)
- [Controllable protein design with particle-based Feynman-Kac steering](https://mlsanity.com/paper/2511.09216/) (2025)
- [Efficient Approximate Posterior Sampling with Annealed Langevin Monte Carlo](https://mlsanity.com/paper/2508.07631/) (2025)

## Built on by (most cited first, 5 of 5)

- [How to Guide Your Flow: Few-Step Alignment via Flow Map Reward Guidance](https://mlsanity.com/paper/2604.27147/) (2026)
- [Are we really tilting? The mechanics of reward guidance in flow and diffusion models](https://mlsanity.com/paper/2606.02884/) (2026)
- [Diffusion-Based Posterior Sampling: A Feynman-Kac Analysis of Bias and Stability](https://mlsanity.com/paper/2605.06538/) (2026)
- [The tractability landscape of diffusion alignment: regularization, rewards, and computational primitives](https://mlsanity.com/paper/2605.11361/) (2026)
- [A Diffusion-Model Subpopulation Digital Twin for Mobile Health Deployment: A Case Study on the HeartSteps Intervention](https://mlsanity.com/paper/2607.21403/) (2026)

## Abstract (arXiv)

Inference-time algorithms are an emerging paradigm in which pre-trained models are used as subroutines to solve downstream tasks. Such algorithms have been proposed for tasks ranging from inverse problems and guided image generation to reasoning. However, the methods currently deployed in practice are heuristics with a variety of failure modes -- and we have very little understanding of when these heuristics can be efficiently improved. In this paper, we consider the task of sampling from a reward-tilted diffusion model -- that is, sampling from pstar(x) propto p(x) exp(r(x)) -- given a reward function r and pre-trained diffusion oracle for p. We provide a fine-grained analysis of the computational tractability of this task for quadratic rewards r(x) = x^top A x + b^top x. We show that linear-reward tilts are always efficiently sampleable -- a simple result that seems to have gone unnoticed in the literature. We use this as a building block, along with a conceptually new ingredient -- the Hubbard-Stratonovich transform -- to provide an efficient algorithm for sampling from low-rank positive-definite quadratic tilts, i.e. r(x) = x^top A x where A is positive-definite and of rank O(1). For negative-definite tilts, i.e. r(x) = - x^top A x where A is positive-definite, we prove that the problem is intractable even if A is of rank 1 (albeit with exponentially-large entries).