🤖 machine learning

DiRecT: Safe Diffusion-Based Planning via Receding-Horizon Denoising

DiRecT is a training-free algorithm that ensures safe planning in diffusion models by enforcing constraints only on the final clean trajectory via receding-horizon denoising, thereby avoiding the over-constraining of intermediate steps that typically degrades sample quality in existing approaches.

Original authors: Paolo Giaretta, Zeyang Li, Navid Azizan

Published 2026-06-16

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Paolo Giaretta, Zeyang Li, Navid Azizan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are teaching a robot to walk through a crowded room to get a cup of coffee. You have a very smart robot that has watched thousands of videos of people doing this. It knows how to move generally, but it doesn't know about the specific chairs or tables in the room right now.

If you just ask the robot to "go get the coffee," it might walk straight into a table because it's trying to be creative. If you try to stop it from hitting the table every single step of the way as it plans, you might end up with a robot that freezes, stutters, or takes a weird, jerky path because you've micromanaged its every thought.

This is the problem the paper DiRecT solves.

The Problem: The "Over-Correction" Trap

The authors explain that existing methods for making AI planners safe are like a nervous parent holding a child's hand too tightly.

The Old Way: As the robot plans its path (step-by-step), the computer checks: "Is this step safe?" If not, it forces the robot to change that specific step immediately.
The Flaw: The robot's planning process involves a lot of "noise" or rough drafts. Checking for safety on these rough drafts is like telling a painter, "Don't make a single brushstroke that looks like a tree," while they are still mixing colors on the palette. It restricts the robot's creativity and often leads to a final result that is ugly, jerky, or fails to reach the goal.

The Solution: DiRecT (The "Receding-Horizon" Coach)

The authors introduce DiRecT, which acts more like a wise coach than a nervous parent.

1. The "Clean Trajectory" Vision
Instead of checking safety on every rough draft, DiRecT waits until the robot has a final, clean plan in mind. It asks: "If you follow this plan exactly, will you hit a table?"

The Analogy: Imagine the robot is drawing a path on a piece of paper. The old methods tried to erase a line the moment the pencil touched the paper if it looked slightly crooked. DiRecT lets the pencil draw freely, then looks at the finished drawing. If the finished drawing hits a table, then it gently nudges the whole drawing to move it away.

2. The "Receding Horizon" Trick
The paper uses a concept called "Receding-Horizon" (inspired by Model Predictive Control).

The Analogy: Think of driving a car at night with foggy headlights. You can only see a few feet ahead.
- Old Method: You try to steer perfectly for the entire 100-mile trip right now, even though you can't see the road.
- DiRecT: You look at the road immediately in front of you, plan a safe path for the next few seconds, drive that, and then re-plan for the next few seconds. You constantly update your plan based on what you see right now, ensuring you never hit a wall, but you don't force the car to drive in a straight line if the road curves.

3. The "Training-Free" Superpower
The best part? DiRecT doesn't need to re-teach the robot. It works with the robot's existing knowledge (the "pre-trained model"). It just adds a safety layer on top of the planning process.

The Analogy: It's like giving a seasoned driver a GPS that warns them of obstacles. You don't need to teach the driver how to drive again; you just give them a tool to avoid the specific potholes in this city.

How It Works in Practice

The paper tested this on several robot tasks:

Maze Navigation: A ball rolling through a maze with new, unexpected walls. DiRecT successfully navigated the maze without hitting walls, while other methods got stuck or crashed.
Robotic Arms: An arm moving to grab an object while avoiding pillars. DiRecT ensured the arm never touched the pillars, even when the pillars were placed in tricky spots.
Multi-Robot Swarms: A group of robots moving together without bumping into each other. DiRecT kept them safe and efficient, even as the number of robots increased.

The Bottom Line

DiRecT is a new way to make AI planners safe. Instead of constantly interrupting the AI's thought process to check for safety (which makes the AI clumsy), it lets the AI think freely and then gently corrects the final plan to ensure it's safe. It's like letting a student write a rough draft without fear, then helping them edit the final version to ensure it's perfect, rather than stopping them after every sentence.

The result is a robot that is both safe (it doesn't crash) and skilled (it moves smoothly and gets the job done).

Technical Summary: DiRecT – Safe Diffusion-Based Planning via Receding-Horizon Denoising

Problem Statement

Diffusion models have emerged as powerful frameworks for planning and control by learning multimodal distributions over actions and trajectories. However, deploying these models in safety-critical tasks faces a significant barrier: reliable inference-time safety enforcement. Existing approaches to constrained diffusion sampling typically enforce feasibility by projecting each noisy intermediate denoising iterate onto the feasible set. This creates a fundamental mismatch: planning constraints are defined on the final clean trajectory, yet intermediate samples are inherently noisy and need not be feasible. Enforcing constraints throughout the sampling process overconstrains the learned reverse dynamics, distorting the generative process and substantially degrading sample quality. Furthermore, projection-based methods often lack a unified mechanism for handling hard constraints alongside additional soft rewards or costs.

Methodology: DiRecT

To address these limitations, the authors introduce DiRecT (Diffusion-based planning via Receding-horizon denoising with Terminal constraints), a training-free algorithm for constrained sampling.

Core Formulation

DiRecT reframes inference-time constraint enforcement through the lens of Stochastic Optimal Control (SOC). The learned reverse diffusion process serves as the nominal stochastic dynamics, while control inputs steer the sampler toward the feasibility of the final clean trajectory. Crucially, constraints are imposed only on the terminal clean sample, avoiding unnecessary restrictions on noisy intermediate iterates. The objective is to find an optimal control drift $u^*$ that minimizes a cost function (balancing terminal cost and control energy) subject to the constraint that the final state $X_0$ lies within a feasible set $S$ .

Algorithmic Derivation

Solving the full terminally constrained SOC problem is computationally intractable due to the high-dimensional, non-convex nature of the neural dynamics and the expectation over all stochastic paths. DiRecT derives a scalable, tractable surrogate using two key approximations inspired by Model Predictive Control (MPC):

Receding-Horizon Surrogate: Instead of solving a full-horizon optimization, DiRecT breaks the problem into a sequence of one-step subproblems. At each denoising step $i$ , it predicts the final clean trajectory implied by the current noisy iterate using Tweedie's formula ( $\hat{x}^\theta_0(X_i, t_i)$ ).
Data-Domain Optimization: To avoid repeatedly solving constrained optimization problems through the neural denoising map, the algorithm approximately inverts Tweedie's formula via a fixed-point argument. This allows the optimization to be moved from the latent space to the data domain. Specifically, the latent displacement is approximated by a scaled difference between predicted clean samples.

The resulting algorithm (Algorithm 1) operates as follows:

Sample a prior noise vector.
Iterate backward from $t_N$ $t_{N}$ to $t_0$ $t_{0}$ :
- Compute an uncontrolled denoising proposal.
- Estimate the corresponding clean sample using the score model.
- Solve a constrained optimization problem in the data domain to find a refined clean sample that satisfies the hard constraints $S$ and minimizes the cost $C$ , while staying close to the uncontrolled prediction.
- Translate this refinement back into a controlled update of the current noisy iterate.
Return the final feasible sample.

This design cleanly separates stochastic denoising from constraint satisfaction, progressively steering samples toward feasible final trajectories without distorting the learned diffusion dynamics.

Key Contributions

Identification of a Key Limitation: The paper identifies that enforcing feasibility on noisy intermediate denoising iterates overconstrains sampling dynamics and degrades sample quality. It proposes formulating constrained diffusion planning as a terminally constrained SOC problem instead.
DiRecT Algorithm: A training-free constrained diffusion sampler that reduces the intractable constrained SOC problem to a sequence of tractable clean-trajectory optimization subproblems. It steers samples toward constraint satisfaction without distorting the learned denoising process.
Flexibility: The method supports off-the-shelf and domain-specific optimizers, handles both equality and inequality constraints, incorporates environment-specific dynamics priors, and optimizes additional soft rewards or costs at inference time.
Empirical Validation: Extensive experiments demonstrate that DiRecT substantially improves deployment safety and task performance over existing diffusion-based planning baselines.

Experimental Results

The authors evaluated DiRecT across four diverse robotic planning benchmarks:

Safe Maze Navigation (Maze2D): In both "broad" and "narrow" maze variants with test-time obstacles, DiRecT achieved a Safety Rate (SR) of 0.970 and 0.940 respectively, with near-zero violations (0.45 and 0.23). In contrast, baselines like Diffuser, Gradient Guidance, and Projected Diffusion achieved SRs ranging from 0.00 to 0.78, with significantly higher violation counts. DiRecT maintained strong task scores while ensuring safety.
Safe Robotic Manipulation (D3IL): In a task requiring a robotic arm to reach a target while avoiding pillars and additional constraints, DiRecT achieved a 100% Safety Rate and 100% Task Success, outperforming all baselines which showed significant drops in safety (e.g., Gradient Guidance at 0.88 SR).
Multi-Robot Motion Planning (MRMP): Tested on environments with up to 20 agents, DiRecT consistently outperformed baselines (including MMD-CBS and Projected Coupled Diffusion) in Constraint Safety and Success Rate, particularly as the number of agents increased. It maintained high adherence to expected motion patterns while satisfying complex collision and velocity constraints.
Contact-Rich Manipulation (PushT): In a coupled generation task involving diverse, non-intersecting trajectories, DiRecT variants (DiRecT-LB and DiRecT-DPP) achieved 100% Constraint Satisfaction while improving diversity (measured by DTW and DFD) and task completion scores compared to other constrained samplers.

Significance and Claims

The paper claims that DiRecT offers a principled approach to safe diffusion-based planning by decoupling the stochastic denoising process from constraint satisfaction. By enforcing constraints only on the final clean sample, it avoids the over-constraining of intermediate dynamics that plagues existing projection-based methods.

The authors emphasize that DiRecT is training-free, making it broadly applicable to existing pretrained diffusion models without the need for retraining or fine-tuning. While this flexibility introduces additional inference-time computation (solving optimization subproblems), the method provides stronger guarantees for safety-critical planning by ensuring hard constraint satisfaction while retaining the flexibility to incorporate soft objectives and domain-specific dynamics. The work highlights the potential for reliable deployment of diffusion models in safety-critical tasks where even a single constraint violation can lead to catastrophic failure.