1. Bibliographic Information

1.1. Title

Planning with Diffusion for Flexible Behavior Synthesis

1.2. Authors

Michael Janner, Yilun Du, Joshua B. Tenenbaum, Sergey Levine. The authors' affiliations are not explicitly stated in the provided text, but Sergey Levine is a prominent researcher in reinforcement learning. Michael Janner and Yilun Du are typically associated with academic research institutions, often under the guidance of senior researchers like Tenenbaum and Levine. Joshua B. Tenenbaum is a well-known professor at MIT, associated with computational cognitive science and AI. Sergey Levine is a professor at UC Berkeley, known for his work in deep reinforcement learning and robotics.

1.3. Journal/Conference

Published at (UTC): 2022-05-20T07:02:03.000Z. While the specific journal or conference is not named in the provided text, the format and content suggest it was published at a top-tier machine learning conference, such as NeurIPS (Advances in Neural Information Processing Systems) or ICLR (International Conference on Learning Representations), given the reference style and the caliber of the authors. These venues are highly reputable and influential in the fields of machine learning and artificial intelligence.

1.4. Publication Year

2022

1.5. Abstract

This paper introduces Diffuser, a novel approach to model-based reinforcement learning (MBRL) that integrates trajectory optimization directly into a diffusion probabilistic model. Traditional MBRL methods often learn an approximate dynamics model and then use classical trajectory optimizers, a combination that frequently exhibits empirical shortcomings. Diffuser aims to unify modeling and planning such that sampling from the model and planning with it become nearly identical processes. The core technical contribution is a diffusion probabilistic model that plans by iteratively denoising trajectories. The authors demonstrate how concepts like classifier-guided sampling and image inpainting can be reinterpreted as effective planning strategies within this framework. The paper explores the unique properties of diffusion-based planning, including its ability to handle long-horizon decision-making and offer test-time flexibility, showcasing its effectiveness in various control settings.

1.6. Original Source Link

https://arxiv.org/abs/2205.09991 PDF Link: https://arxiv.org/pdf/2205.09991v2.pdf Publication Status: Preprint (arXiv) with an indicated publication date. While arXiv hosts preprints, many papers first appear there before formal publication in conferences or journals.

2. Executive Summary

2.1. Background & Motivation

The core problem the paper addresses is the empirical shortcomings of combining learned approximate dynamics models with classical trajectory optimizers in model-based reinforcement learning (MBRL). While conceptually simple—learning dynamics and then optimizing trajectories—this approach often leads to plans that resemble adversarial examples rather than optimal trajectories, because learned models may not be well-suited to standard trajectory optimization. This suggests a disconnect between the model learning and the planning processes.

This problem is important because MBRL promises greater sample efficiency and the ability to plan for novel situations compared to model-free methods. However, the current challenges often force MBRL algorithms to borrow heavily from model-free techniques or resort to simpler, gradient-free trajectory optimization routines to mitigate these issues. The specific challenge is that learned models (which are inherently approximations) can be exploited by trajectory optimizers (which are designed for ground-truth dynamics), leading to unstable and suboptimal behaviors.

The paper's entry point is to reconsider the relationship between modeling and planning. Instead of learning a dynamics model and then plugging it into an external optimizer, the authors propose to "fold as much of the trajectory optimization pipeline as possible into the modeling problem." The innovative idea is to design a model that is inherently amenable to trajectory optimization, making sampling from the model and planning with it almost the same process. This requires a shift in model design, focusing on properties like long-horizon accuracy and action distributions, while remaining reward-agnostic to support multi-task planning.

2.2. Main Contributions / Findings

The paper's primary contributions are:

• Novel Diffusion-based Planning Framework (Diffuser): Proposing Diffuser, a denoising diffusion probabilistic model specifically designed for trajectory data. This model directly learns to generate entire trajectories (sequences of states and actions) rather than just single-step dynamics predictions.

• Unification of Sampling and Planning: Developing a framework where planning becomes nearly identical to sampling from the Diffuser model, guided by auxiliary perturbation functions. This tighter coupling addresses the adversarial exploitation issues of traditional MBRL.

• Reinterpretation of Existing Techniques: Demonstrating how classifier-guided sampling (for maximizing rewards) and image inpainting (for satisfying constraints like start/goal states) can be effectively reinterpreted as coherent planning strategies within the diffusion framework for reinforcement learning.

• Identification of Key Properties: Exploring and highlighting several unique and useful properties of diffusion-based planning:

  • Long-horizon scalability: Diffuser is trained for trajectory accuracy, avoiding compounding single-step errors.
  • Task compositionality: The model is reward-agnostic, allowing for planning with new or combined reward functions at test time.
  • Temporal compositionality: It generates globally coherent trajectories by iteratively improving local consistency, enabling generalization by stitching together in-distribution subsequences.
  • Effective non-greedy planning: The training procedure improves planning capabilities, allowing it to solve long-horizon, sparse-reward problems.

• Empirical Validation: Demonstrating the effectiveness of Diffuser in control settings that demand long-horizon decision-making (e.g., Maze2D), test-time flexibility (e.g., block stacking with novel goals), and offline reinforcement learning (e.g., D4RL locomotion), where it substantially outperforms or is comparable to prior state-of-the-art methods.

  The key conclusion is that designing a generative model that inherently supports planning, rather than merely predicting dynamics, can lead to more robust and flexible reinforcement learning agents, particularly for challenging long-horizon and multi-task problems. These findings solve the problem of learned models being poorly suited for conventional trajectory optimizers by integrating the planning process directly into the generative model itself.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

To understand this paper, a beginner should grasp the following fundamental concepts:

• Reinforcement Learning (RL): An area of machine learning concerned with how intelligent agents ought to take actions in an environment to maximize the notion of cumulative reward. The agent learns an optimal policy (a mapping from states to actions) through trial and error.
• Model-Based Reinforcement Learning (MBRL): A subfield of RL where the agent learns or is provided with a model of the environment's dynamics. This model predicts how the environment will change in response to actions and what rewards will be received. The agent can then use this model to plan future actions without physically interacting with the environment, often leading to better sample efficiency.
• Dynamics Model: In MBRL, this is a learned approximation of how the environment transitions from one state to another given an action. Mathematically, it predicts $s_{t+1} = f(s_t, a_t)$, where $s_t$ is the state at time $t$ and $a_t$ is the action taken.
• Trajectory Optimization: A classical control technique used to find a sequence of actions (a trajectory) that optimizes a given objective function (e.g., minimizes cost or maximizes reward) subject to system dynamics and constraints. Methods like Iterative Linear Quadratic Regulator (iLQR) or Model Predictive Path Integral (MPPI) are examples.
• Generative Models: A class of statistical models that learn the underlying distribution of data and can then generate new data samples that resemble the training data. Examples include Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and Diffusion Models.
• Diffusion Probabilistic Models (DPMs): A specific type of generative model that works by systematically destroying training data through the successive addition of Gaussian noise (the forward diffusion process) and then learning to reverse this process, i.e., to denoise the data to recover original samples (the reverse diffusion process). They have shown great success in image generation.
• Trajectory: A sequence of states and actions over a period, e.g., $\boldsymbol{\tau} = (s_0, a_0, s_1, a_1, \ldots, s_T, a_T)$, where $T$ is the planning horizon.
• Receding-Horizon Control (Model Predictive Control - MPC): A control strategy where at each time step, an optimization problem is solved over a finite future horizon to determine a sequence of actions. Only the first action from this sequence is executed, and then the process is repeated from the new state. This provides robustness to model inaccuracies and unexpected disturbances.
• Sparse Rewards: A common challenge in RL where the agent receives meaningful reward signals only very rarely (e.g., a reward of 1 only upon reaching a distant goal, and 0 otherwise). This makes learning difficult as the agent struggles to identify which actions led to the reward.
• Goal-Conditioned Reinforcement Learning: A type of RL where the agent's policy is conditioned on a specific goal state, allowing it to learn to achieve various goals within the same environment.

3.2. Previous Works

The paper builds upon and differentiates itself from several lines of prior work:

• Classical Trajectory Optimization (e.g., Tassa et al., 2012; Posa et al., 2014; Kelly, 2017): These methods are well-understood for systems with known dynamics. The paper mentions that iLQR (Iterative Linear Quadratic Regulator) and DDP (Differential Dynamic Programming) are powerful trajectory optimizers. The issue arises when plugging in learned models, as these optimizers can find "adversarial examples" in the learned model that don't correspond to physical reality, leading to poor performance (Talvitie, 2014; Ke et al., 2018).
• Model-Based RL with Simple Gradient-Free Optimizers (e.g., Nagabandi et al., 2018; Botev et al., 2013; Chua et al., 2018): To avoid the issues with complex optimizers exploiting learned models, some MBRL methods use simpler planning routines like random shooting (generating many random action sequences and picking the best) or the Cross-Entropy Method (CEM) (iteratively refining a distribution over action sequences). While more robust, these are less powerful than gradient-based optimizers.
• Deep Generative Models in MBRL: Recent advances have brought deep generative models into dynamics modeling using various architectures:
  • Convolutional U-networks (Kaiser et al., 2020)
  • Stochastic recurrent networks (Ke et al., 2018; Hafner et al., 2021a; Ha & Schmidhuber, 2018)
  • Vector-quantized autoencoders (Hafner et al., 2021b; Ozair et al., 2021)
  • Neural ODEs (Du et al., 2020a)
  • Normalizing flows (Rinehart et al., 2020; Janner et al., 2020)
  • Generative Adversarial Networks (GANs) (Eysenbach et al., 2021)
  • Energy-Based Models (EBMs) (Du et al., 2019)
  • Graph Neural Networks (Sanchez-Gonzalez et al., 2018)
  • Neural Radiance Fields (Li et al., 2021)
  • Transformers (Janner et al., 2021; Chen et al., 2021a) These models primarily focus on learning accurate environment dynamics, maintaining an "abstraction barrier" between the model and the planner.
• Non-autoregressive Trajectory-level Dynamics Models (Lambert et al., 2020): These works explore predicting entire trajectories non-autoregressively for long-horizon prediction, but still typically separate the modeling from the planning process.
• Breaking the Abstraction Barrier (Different Ways):
  • Autoregressive latent-space models for reward prediction (Tamar et al., 2016; Oh et al., 2017; Schrittwieser et al., 2019): These learn models that are aware of rewards or values.
  • Value-aware loss functions for dynamics models (Farahmand et al., 2017): Modifying dynamics model training to prioritize regions important for value.
  • Collocation techniques with learned single-step energies (Du et al., 2019; Rybkin et al., 2021): Using learned energy functions to find trajectories.
• Diffusion Models in Generative AI (Sohl-Dickstein et al., 2015; Ho et al., 2020): This paper specifically leverages Diffusion Probabilistic Models, which have recently gained prominence in image synthesis (Song et al., 2021; Dhariwal & Nichol, 2021), waveforms (Chen et al., 2021c), 3D shapes (Zhou et al., 2021), and text (Austin et al., 2021). The iterative denoising process of DPMs, particularly with classifier-guided sampling (Dhariwal & Nichol, 2021) and compositionality (Du et al., 2020b), is central to Diffuser's planning mechanism.

Diffusion Probabilistic Models (Ho et al., 2020; Sohl-Dickstein et al., 2015)

Diffusion models are generative models that learn to reverse a gradual noisy process. The forward diffusion process $q(\tau^i \mid \tau^{i-1})$ gradually adds noise to a data sample $\tau^0$ over $N$ steps, transforming it into a latent variable $\tau^N$ that is approximately a standard Gaussian distribution. The reverse diffusion process $p_\theta(\tau^{i-1} \mid \tau^i)$ learns to denoise from $\tau^N$ back to $\tau^0$. This is parameterized by a neural network $\theta$.

The data distribution induced by the model is given by: $ p_{\theta}(\tau^{0}) = \int p(\tau^{N})\prod_{i = 1}^{N}p_{\theta}(\tau^{i - 1}\mid \tau^{i})\mathrm{d}\tau^{1:N} $ where $p(\tau^N)$ is a standard Gaussian prior (e.g., $\mathcal{N}(0, I)$) and $\tau^0$ denotes the original (noiseless) data. The parameters $\theta$ are optimized by minimizing a variational bound on the negative log-likelihood of the reverse process: $ \theta^{*} = \arg \min_{\theta} - \mathbb{E}{\tau^{0}}[\log p{\theta}(\tau^{0})] $ The reverse process transitions are often parameterized as Gaussian: $ p_{\theta}(\tau^{i - 1}\mid \tau^{i}) = \mathcal{N}(\tau^{i - 1}\mid \mu_{\theta}(\tau^{i},i),\Sigma^{i}) $ where $\mu_{\theta}(\tau^{i},i)$ is the predicted mean and $\Sigma^{i}$ are fixed, timestep-dependent covariances. The forward process $q(\tau^i \mid \tau^{i-1})$ is typically prespecified (e.g., fixed variance Gaussian noise). The model learns to predict the noise $\epsilon_\theta(\tau^i, i)$ added at each step, from which the mean $\mu_\theta$ can be derived.

3.3. Technological Evolution

The evolution of model-based reinforcement learning has moved from strictly separating dynamics modeling from planning (where dynamics models are essentially proxies for the environment) towards more integrated approaches. Initially, MBRL focused on learning accurate single-step dynamics models and then using powerful, often gradient-based, classical trajectory optimizers. However, this often led to "adversarial examples" where the optimizers exploited imperfections in the learned models.

To counteract this, many MBRL methods simplified the planning component, resorting to gradient-free methods like random shooting or CEM, effectively sacrificing optimality for robustness. Concurrently, advancements in deep generative models opened new avenues for more expressive dynamics models. This paper represents a further evolution, moving beyond just accurate dynamics prediction to designing a generative model (Diffuser) whose sampling process is the planning process. This tightly couples the two, aiming to resolve the long-standing tension between learned dynamics models and classical trajectory optimizers by making the model itself a "learned planner." It integrates the flexibility of diffusion models (like guided sampling and inpainting) directly into the decision-making loop, allowing for robust long-horizon planning and test-time adaptability.

3.4. Differentiation Analysis

Compared to the main methods in related work, the core differences and innovations of Diffuser are:

• Unified Modeling and Planning: Unlike traditional MBRL where a learned dynamics model is merely "plugged into" a classical trajectory optimizer, Diffuser integrates the planning logic directly into the generative model's sampling process. This means sampling from Diffuser is planning.

• Non-Autoregressive Trajectory Generation: Most dynamics models, even deep generative ones, predict states and actions autoregressively (one step at a time). Diffuser generates entire trajectories (sequences of states and actions over a horizon) non-autoregressively and concurrently, which is crucial for handling anti-causal dependencies in decision-making (where future goals influence present actions).

• Diffusion Model Paradigm: Diffuser is the first to apply denoising diffusion probabilistic models to trajectory generation for reinforcement learning. This leverages the iterative refinement and flexible conditioning capabilities of diffusion models, which are novel for planning.

• Flexible Conditioning for Planning: The diffusion framework naturally allows for classifier-guided sampling (to maximize rewards) and inpainting (to satisfy constraints like start/goal states). This inherent flexibility for specifying objectives and constraints at test time is a significant advantage over models that are rigidly trained for specific reward functions or tasks.

• Robustness to Learned Model Exploitation: By blurring the line between model and planner, Diffuser inherently designs the model to be robust for planning. The iterative denoising process, guided by a perturbation function, encourages physically realistic and high-reward trajectories, circumventing the issue of optimizers finding adversarial examples in approximate dynamics models.

• Temporal and Task Compositionality: Diffuser demonstrates an ability to generalize by "stitching together" familiar subsequences from training data to create novel, globally coherent trajectories (temporal compositionality). Furthermore, it can be guided by new reward functions unseen during training (task compositionality) due to its reward-agnostic prior over trajectories.

  In essence, Diffuser shifts the paradigm from "learn a model, then plan" to "learn a planner that is a model," leveraging the unique strengths of diffusion probabilistic models for robust, flexible, and long-horizon decision-making.

4. Methodology

4.1. Principles

The core idea behind Diffuser is to replace the traditional separation of dynamics modeling and trajectory optimization with a unified generative modeling approach. Instead of learning a single-step dynamics model and then applying a separate optimizer, Diffuser learns a generative model of entire trajectories. The planning process then becomes nearly identical to sampling from this generative model, but with an added guidance mechanism that biases the sampling towards trajectories that satisfy certain objectives (like maximizing reward) or constraints (like starting at a specific state or reaching a goal).

The theoretical basis comes from Diffusion Probabilistic Models (DPMs), which are excellent at learning complex data distributions and generating high-quality samples through an iterative denoising process. The key intuition is that by learning to denoise trajectories (sequences of states and actions) rather than just static images or single-step dynamics, the model implicitly learns physically plausible behaviors. Then, by introducing a perturbation function $h(\tau)$ that encodes rewards or constraints, the denoising process can be steered to generate optimal or feasible plans. This approach leverages the flexible conditioning capabilities of DPMs, where classifier guidance (for rewards) and inpainting (for constraints) become natural planning strategies. This tight coupling ensures that the model's learned properties (like long-horizon consistency) directly translate into effective planning capabilities.

4.2. Core Methodology In-depth (Layer by Layer)

Diffuser instantiates the idea of tightly coupling modeling and planning through a trajectory-level diffusion probabilistic model.

4.2.1. Trajectory Representation

For Diffuser to model and plan, it needs a suitable representation for trajectories. Unlike typical dynamics models that might predict the next state or action given the current, Diffuser predicts all timesteps of a plan simultaneously. Since the effectiveness of the controller is as important as state predictions, states and actions are predicted jointly. Actions are treated as additional dimensions of the state for prediction purposes.

The trajectory $\boldsymbol{\tau}$ is represented as a two-dimensional array: $ \boldsymbol {\tau} = \begin{bmatrix} \mathbf{s}_0 & \mathbf{s}_1 & \dots & \mathbf{s}_T \ \mathbf{a}_0 & \mathbf{a}_1 & \dots & \mathbf{a}_T \end{bmatrix} \quad (2) $ where:

• $\mathbf{s}_t$ is the state vector at timestep $t$.

• $\mathbf{a}_t$ is the action vector at timestep $t$.

• $T$ is the planning horizon, representing the total number of timesteps in the trajectory.

• Each column represents a state-action pair at a specific timestep. The array has two rows (one for states, one for actions) and $T+1$ columns (for timesteps 0 to $T$).

  This representation allows the model to process both state and action information cohesively across the entire planning horizon.

4.2.2. Architecture

Diffuser's architecture is designed to satisfy three key criteria for trajectory planning:

1. Non-autoregressive Prediction: The entire trajectory should be predicted concurrently, not one timestep at a time. This addresses the "anti-causal" nature of decision-making, where future goals can influence present actions.

2. Temporal Locality: Each step of the denoising process should primarily rely on nearby timesteps (both past and future) to enforce local consistency. Global coherence then emerges from composing many such local denoising steps.

3. Equivariance: The model should be equivariant along the planning horizon dimension (meaning it can handle variable-length trajectories) but not the state and action features dimension.

   These criteria are met using a U-Net like architecture (common in image-based diffusion models) but adapted for temporal data. Instead of two-dimensional spatial convolutions, it uses one-dimensional temporal convolutions.

The architecture (Figure A1) consists of a U-Net structure with 6 repeated residual blocks. Each residual block further consists of:

• Two temporal convolutions: These are 1D convolutions applied along the time dimension of the trajectory array. They allow the model to capture dependencies between adjacent state-action pairs.

• Group normalization (GN) (Wu & He, 2018): A normalization technique applied after convolutions, which helps stabilize training.

• Mish nonlinearity (Misra, 2019): An activation function applied after normalization, introducing non-linearity to the network.

• Timestep embeddings: These are produced by a single fully-connected layer and are added to the activations of the first temporal convolution within each block. This allows the model to be aware of the current diffusion timestep $i$.

  Because the model is fully convolutional, its planning horizon $T$ is not fixed by the architecture but by the input dimensionality of the noise, allowing for variable-length plans.

The following figure (Figure A1 from the original paper) shows the system architecture:

该图像是一个示意图，展示了一个包含卷积层和全连接层的深度学习模型结构。左侧为模型的总体架构，右侧为网络的细节部分，其中包括时间 t 和输入 X 的处理流程，以及 GN Mish 激活函数的使用。

4.2.3. Training

Diffuser is trained to parameterize a learned gradient $\epsilon_{\theta}(\tau^{i}, i)$ of the trajectory denoising process. This is a common approach in diffusion models, where the model learns to predict the noise component $\epsilon$ that was added to a clean data sample $\tau^0$ to obtain a noisy sample $\tau^i$. From this predicted noise, the mean $\mu_{\theta}$ of the reverse process can be solved in closed form (Ho et al., 2020).

The simplified objective for training the $\epsilon$-model is given by: $ \mathcal{L}(\theta) = \mathbb{E}{i,\epsilon ,\tau^0}\left[|\epsilon -\epsilon\theta (\boldsymbol{\tau}^i,i)| ^2\right] $ where:

• $i \sim \mathcal{U}\{1,2, \dots, N\}$: The diffusion timestep, uniformly sampled from 1 to $N$. $N$ is the total number of diffusion steps.

• $\epsilon \sim \mathcal{N}(0, I)$: The noise target, which is standard Gaussian noise added to the clean trajectory.

• $\tau^0$: The original (noiseless) trajectory from the training dataset.

• $\tau^i$: The trajectory $\tau^0$ corrupted with noise $\epsilon$ at diffusion timestep $i$. This is obtained by applying the forward diffusion process for $i$ steps.

• $\epsilon_\theta (\boldsymbol{\tau}^i,i)$: The Diffuser model's prediction of the noise component that was added to $\tau^0$ to produce $\tau^i$.

• $\|\cdot\|^2$: The squared $L_2$ norm, indicating that the model is trained to minimize the difference between the predicted noise and the actual noise added.

  The reverse process covariances $\Sigma^i$ are not learned but follow a fixed cosine schedule (Nichol & Dhariwal, 2021), a common choice in diffusion models for stable training.

4.2.4. Reinforcement Learning as Guided Sampling

To apply Diffuser to reinforcement learning, the concept of reward must be incorporated. This is achieved by reinterpreting RL as a conditional sampling problem, inspired by the control-as-inference framework (Levine, 2018).

The idea is to sample trajectories from a perturbed distribution $\tilde{p}_{\theta}(\tau)$ that is proportional to the original Diffuser distribution $p_{\theta}(\tau)$ weighted by a perturbation function $h(\tau)$. $ \tilde{p}{\theta}(\tau)\propto p{\theta}(\tau)h(\tau) \quad (1) $ For reinforcement learning, $h(\tau)$ is defined based on the optimality of a trajectory. Let $\mathcal{O}_t$ be a binary random variable indicating the optimality of timestep $t$, with p(\mathcal{O}_t = 1) = \exp (r(\mathbf{s}_t,\mathbf{a}_t)), where $r(\mathbf{s}_t,\mathbf{a}_t)$ is the reward at state $\mathbf{s}_t$ and action $\mathbf{a}_t$. Then, $h(\tau)$ can represent the probability of the entire trajectory being optimal: $h(\tau) = p(\mathcal{O}_{1:T}\mid \tau)$. The perturbed distribution becomes: $ \tilde{p}{\theta}(\pmb {\tau}) = p(\pmb {\tau}|\mathcal{O}{1:T} = 1)\propto p(\pmb {\tau})p(\mathcal{O}_{1:T} = 1|\pmb {\tau}) $ This means we are looking for trajectories that are both physically realistic (high $p_{\theta}(\tau)$) and high-reward (high $p(\mathcal{O}_{1:T} = 1|\pmb {\tau})$).

While exact sampling from this perturbed distribution is intractable, it can be approximated by modifying the reverse diffusion process transitions, especially if $p(\mathcal{O}_{1:T} \mid \tau^i)$ is sufficiently smooth. The modified reverse transition mean becomes: $ p_{\theta}(\tau^{i - 1} \mid \tau^i, \mathcal{O}_{1:T}) \approx \mathcal{N}(\tau^{i - 1}; \mu + \Sigma g, \Sigma) \quad (3) $ where:

• $\mu, \Sigma$: The parameters (mean and covariance) of the original reverse process transition $p_{\theta}(\tau^{i - 1} \mid \tau^i)$.
• $g$: A gradient term that guides the sampling towards high-reward trajectories, analogous to classifier-guided sampling in image generation. This gradient is calculated as: $ g = \nabla_{\tau}\log p(\mathcal{O}{1:T}\mid \tau) |{\tau = \mu} $ Since p(\mathcal{O}_{1:T} = 1|\pmb {\tau}) = \prod_{t=0}^T p(\mathcal{O}_t=1|\mathbf{s}_t, \mathbf{a}_t) = \prod_{t=0}^T \exp(r(\mathbf{s}_t, \mathbf{a}_t)) = \exp(\sum_{t=0}^T r(\mathbf{s}_t, \mathbf{a}_t)) = \exp(\mathcal{J}(\tau)), where $\mathcal{J}(\tau)$ is the cumulative reward (return) of the trajectory $\tau$. The gradient $g$ simplifies to: $ g = \sum_{t = 0}^{T}\nabla_{\mathbf{s}_t,\mathbf{a}_t}r(\mathbf{s}_t,\mathbf{a}t)|{(\mathbf{s}_t,\mathbf{a}_t) = \mu_t} = \nabla \mathcal{J}(\mu) $ This means the guide $g$ is the gradient of the total return $\mathcal{J}(\tau)$ with respect to the trajectory $\tau$ (evaluated at the current estimated mean $\mu$).

The guided sampling procedure then involves:

1. Training Diffuser $p_{\theta}(\tau)$ on all available trajectory data.

2. Training a separate reward predictor model $\mathcal{J}_{\phi}$ to estimate the cumulative rewards (returns) of trajectory samples $\tau^i$.

3. During sampling, the gradients of $\mathcal{J}_{\phi}$ are used to guide the denoising process, modifying the mean $\mu$ of the reverse transitions by adding $\alpha \Sigma \nabla \mathcal{J}(\mu)$, where $\alpha$ is a guide scale hyperparameter.

   The overall planning strategy uses a receding-horizon control loop: after sampling a trajectory $\tau$, the first action $\tau_{\mathbf{a}_0}^0$ is executed in the environment, and the planning process restarts from the new state.

4.2.5. Goal-Conditioned RL as Inpainting

For planning problems that are more about constraint satisfaction (e.g., reaching a specific goal location) than pure reward maximization, Diffuser reinterprets them as an inpainting problem. This is possible due to the two-dimensional array representation of trajectories. Constraints on specific states or actions at particular timesteps are treated like observed pixels in an image, and the diffusion model "inpaints" the unobserved parts of the trajectory in a manner consistent with these constraints.

For a state constraint $\mathbf{c}_t$ at timestep $t$, the perturbation function $h(\boldsymbol{\tau})$ is a Dirac delta function: $ h(\pmb {\tau}) = \delta_{\mathbf{c}_t}(\mathbf{s}_0,\mathbf{a}_0,\dots,\mathbf{s}_T,\mathbf{a}_T) = \begin{cases} +\infty & \mathrm{if}\mathbf{c}_t = \mathbf{s}_t\ 0 & \mathrm{otherwise} \end{cases} $ A similar definition applies to action constraints. In practice, this is implemented by:

1. Sampling from the unperturbed reverse process $\tau^{i - 1} \sim p_{\theta}(\tau^{i - 1} \mid \tau^i)$.

2. Immediately replacing the sampled values at the constrained locations with the conditioning values $\mathbf{c}_t$ after each diffusion timestep $i \in \{0, 1, \ldots, N\}$. This effectively forces the trajectory to adhere to the constraints throughout the denoising process.

   Crucially, even reward maximization problems use inpainting to enforce the starting state. All sampled trajectories must begin at the current state of the environment. This is implemented in Algorithm 1, line 10, by setting the first state of the plan $\pmb{\tau}_{\mathbf{s}_0}^{i}$ to the observed state $\mathbf{s}$.

The following is the pseudocode for the guided planning method (Algorithm 1 from the original paper):

## Algorithm 1 Guided Diffusion Planning  

1: Require Diffuser $\mu_{\theta}$ guide $\mathcal{J}$ scale $\alpha$ , covariances $\Sigma^{i}$ 
2: while not done do 
3: Observe state $\mathbf{s}$ , initialize plan $\tau^{N} \sim \mathcal{N}(0, I)$ 
4: for $i = N, \dots , 1$ do 
5: // parameters of reverse transition 
6: $\mu \leftarrow \mu_{\theta}(\pmb{\tau}^{i})$ 
7: // guide using gradients of return 
8: $\pmb{\tau}^{i - 1} \sim \mathcal{N}(\mu + \alpha \Sigma \nabla \mathcal{J}(\mu), \Sigma^{i})$ 
9: // constrain first state of plan 
10: $\pmb{\tau}_{\mathbf{s}_0}^{i} \leftarrow \pmb{s}$ 
11: Execute first action of plan $\tau_{\mathbf{a}_0}^{0}$ 

Explanation of Algorithm 1:

• Line 1: Requires the trained Diffuser model (specifically, its mean prediction function $\mu_{\theta}$), a trained guide function $\mathcal{J}$ (the reward predictor), a guide scale $\alpha$, and the fixed covariances $\Sigma^{i}$ for the reverse diffusion process.
• Line 2: The while not done do loop represents the receding-horizon control loop, where planning and execution continue until the task is complete.
• Line 3: At the beginning of each planning cycle, the current state $\mathbf{s}$ is observed from the environment. A noisy initial plan $\tau^{N}$ is sampled from a standard Gaussian distribution $\mathcal{N}(0, I)$. This is the starting point for the denoising process.
• Line 4: The for loop iterates through the diffusion timesteps from $N$ down to 1, performing the iterative denoising.
• Line 6: The Diffuser model predicts the mean $\mu$ of the reverse transition, given the current noisy trajectory $\tau^i$.
• Line 8: This is the core guided sampling step. The next less noisy trajectory $\tau^{i-1}$ is sampled from a Gaussian distribution. Its mean is modified by adding a guidance term $\alpha \Sigma \nabla \mathcal{J}(\mu)$ to the model's predicted mean $\mu$. $\nabla \mathcal{J}(\mu)$ is the gradient of the return prediction with respect to the trajectory, which steers the sampling towards higher-reward trajectories.
• Line 10: This line implements the inpainting mechanism for state conditioning. It ensures that the first state ($\mathbf{s}_0$) of the plan $\tau^i$ is always forced to be the observed current state $\mathbf{s}$. This is crucial for grounding the plan in the real environment.
• Line 11: After the denoising process completes (i.e., $\tau^0$ is generated), the first action of the resulting plan $\tau_{\mathbf{a}_0}^{0}$ is executed in the environment. The loop then repeats with the new observed state.

4.2.6. Properties of Diffuser Planners

The paper highlights several important properties of Diffuser that distinguish it from standard dynamics models and non-autoregressive trajectory prediction:

• Learned long-horizon planning: Diffuser's planning procedure is inherently linked to its ability to predict accurate long-horizon trajectories. This allows it to generate feasible plans even in sparse reward settings where traditional shooting-based approaches struggle (Figure 3a). The following figure (Figure 3 from the original paper) illustrates the properties of diffusion planners:

  该图像是一个示意图，展示了扩散模型在轨迹优化中的应用。图中通过去噪过程展示了从噪声到清晰轨迹的转变，以及基于数据和计划结果的对比，强调了奖励函数和计划之间的关系。

• Temporal compositionality: Diffuser generates globally coherent trajectories by iteratively improving local consistency (as described in Section 3.1, Figure 2). This means it can "stitch together" familiar subsequences from its training data in novel ways to generalize to new, unseen trajectories (Figure 3b). For example, if trained on straight-line movements, it can compose them to form V-shaped paths. The following figure (Figure 2 from the original paper) visualizes the denoising process:

  该图像是一个示意图，展示了在规划过程中使用的扩散器结构包含的状态和动作序列，以及局部感受野、去噪和规划视野的关系。该模型通过去噪处理逐步优化轨迹，同时将反映在局域感受野内的多个状态和动作进行整合，以实现有效规划。

• Variable-length plans: Because the model is fully convolutional, the planning horizon $T$ is not a fixed architectural choice. It is determined by the size of the input noise $\tau^N \sim \mathcal{N}(0, \mathbf{I})$ used to initialize the denoising process. This allows for dynamic adjustment of plan length (Figure 3c).

• Task compositionality: Diffuser learns a prior over plausible behaviors that is independent of the reward function. This allows for planning with new, even unseen, reward functions at test time by simply modifying the perturbation function $h(\tau)$ (or combining multiple perturbations). This is demonstrated by planning for a new reward function after training (Figure 3d).

5. Experimental Setup

The experimental evaluation focuses on three key capabilities of Diffuser: (1) long-horizon planning without manual reward shaping, (2) generalization to new goal configurations, and (3) recovery of effective controllers from heterogeneous, varying-quality data. The section concludes with practical runtime considerations.

5.1. Datasets

5.1.1. Maze2D Environments

• Source & Characteristics: Maze2D environments (Fu et al., 2020) are grid-world-like environments where an agent must navigate from a starting point to a goal location. These environments are characterized by sparse rewards: a reward of 1 is given only upon reaching the goal, and 0 otherwise. This sparsity makes them challenging for long-horizon planning, as it can take hundreds of steps to reach the goal.

• Data Sample: The training data for Maze2D is described as "undirected," meaning it consists of an expert controller navigating to and from randomly selected locations. This provides varied trajectories but not necessarily goal-directed ones.

• Choice Justification: These environments are chosen specifically to test long-horizon planning capabilities due to their sparse reward structure, where credit assignment is difficult for many model-free algorithms.

• Multi2D Variant: A multi-task variant called Multi2D is also used, where goal locations are randomized at the beginning of each episode. This tests the ability to generalize to new goal configurations unseen during training.

  The following figure (Figure 4 from the original paper) visualizes planning in Maze2D:

  该图像是一个示意图，展示了在不同规模（U-Maze，Medium，Large）环境中的轨迹去噪过程。图中使用条形向右指示去噪进程，展示了从噪声到清晰轨迹的转变，说明了扩散模型在轨迹优化中的应用。

5.1.2. Block Stacking Tasks

• Source & Characteristics: A suite of block stacking tasks is constructed. These tasks involve manipulating blocks to form specific arrangements. Training data consists of 10,000 trajectories generated by PDDLStream (Garrett et al., 2020), a symbolic planning system. Rewards are 1 for successful stack placements and 0 otherwise.

• Three Settings:

  1. Unconditional Stacking: Build the tallest possible block tower.
  2. Conditional Stacking: Build a tower with a specified order of blocks.
  3. Rearrangement: Match a novel arrangement of reference blocks' locations.

• Choice Justification: These tasks are challenging diagnostics of test-time flexibility. They require the controller to venture into novel states not present in the training data when executing partial stacks for randomized goals, testing generalization and adaptability.

• Data Sample: The image below (Figure 5 from the original paper) provides a visual example of a block stacking sequence, showing the state changes.

  该图像是一个示意图，展示了通过扩散模型进行行为合成的过程。从左侧的原始物体到右侧的变换过程，图中展示了不同的状态和位置，逐步过渡到目标构型，强调了在控制任务中长期决策的灵活性。

5.1.3. D4RL Offline Locomotion Suite

• Source & Characteristics: The D4RL (Datasets for Deep Data-Driven Reinforcement Learning) offline locomotion suite (Fu et al., 2020) is a collection of benchmark datasets for offline RL. These datasets contain heterogeneous trajectories of varying quality (e.g., expert, medium, medium-replay, random) for continuous control tasks like HalfCheetah, Hopper, and Walker2d.
• Choice Justification: This suite is used to evaluate Diffuser's capacity to learn an effective single-task controller from diverse offline data, which is a key challenge in offline RL.

5.2. Evaluation Metrics

The paper uses the normalized score as the primary evaluation metric across all environments.

Normalized Score

• Conceptual Definition: This metric quantifies the performance of an agent relative to a human expert and a random policy. A score of 100 typically represents the performance of an expert policy (or the best possible performance for the task), while 0 represents a random policy. This allows for standardized comparison across different tasks and environments.
• Mathematical Formula: The paper does not explicitly provide the formula for the normalized score, but it is a standard metric in the D4RL benchmark. It is typically calculated as: $ \text{Normalized Score} = 100 \times \frac{\text{Agent's Return} - \text{Random Policy Return}}{\text{Expert Policy Return} - \text{Random Policy Return}} $
• Symbol Explanation:
  • $\text{Agent's Return}$: The cumulative reward achieved by the evaluated agent.
  • $\text{Random Policy Return}$: The cumulative reward achieved by a baseline random policy in the same environment.
  • $\text{Expert Policy Return}$: The cumulative reward achieved by a human expert or a highly optimized policy.

5.3. Baselines

The paper compares Diffuser against a comprehensive set of baseline algorithms, categorized by their approach:

5.3.1. Maze2D Environments

• MPPI (Model Predictive Path Integral): A classical trajectory optimization method that uses sampling to evaluate candidate action sequences. In this context, it uses the ground-truth dynamics.
• CQL (Conservative Q-Learning): A model-free offline reinforcement learning algorithm (Kumar et al., 2020).
• IQL (Implicit Q-Learning): Another model-free offline reinforcement learning algorithm (Kostrikov et al., 2022).
• Goal-conditioned IQL with Hindsight Experience Relabeling: For the Multi2D multi-task setting, IQL is adapted to be goal-conditioned and uses hindsight experience relabeling (Andrychowicz et al., 2017) to select goals during training.

5.3.2. Block Stacking Tasks

• BCQ (Batch-Constrained Q-learning): A model-free offline reinforcement learning algorithm (Fujimoto et al., 2019).
• CQL (Conservative Q-Learning): (Kumar et al., 2020). For conditional and rearrangement tasks, goal-conditioned variants are used.

5.3.3. D4RL Offline Locomotion Suite

• Model-Free RL Algorithms:
  • CQL (Conservative Q-Learning): (Kumar et al., 2020).
  • IQL (Implicit Q-Learning): (Kostrikov et al., 2022).
  • BC (Behavioral Cloning): A simple baseline that learns a policy by directly imitating expert actions from the dataset.
• Return-Conditioning Approaches:
  • DT (Decision Transformer): Models RL as a sequence modeling problem, predicting actions by conditioning on desired future returns (Chen et al., 2021b).
• Model-Based RL Approaches:

  • TT (Trajectory Transformer): A model-based approach that also uses sequence modeling to predict trajectories (Janner et al., 2021).

  • MOPO (Model-based Offline Policy Optimization): An offline MBRL method that uses a learned dynamics model to generate synthetic data for policy training (Yu et al., 2020).

  • MOREL (Model-based Offline Reinforcement Learning): Another offline MBRL method (Kidambi et al., 2020).

  • MBOP (Model-Based Offline Planning): An offline MBRL approach that explicitly performs planning with a learned model (Argenson & Dulac-Arnold, 2021).

    These baselines represent state-of-the-art or standard approaches across model-free, return-conditioning, and model-based offline reinforcement learning, making them representative for comparison.

6. Results & Analysis

6.1. Core Results Analysis

The experimental results strongly validate Diffuser's effectiveness, particularly in settings that require long-horizon reasoning and test-time flexibility.

6.1.1. Long Horizon Multi-Task Planning (Maze2D)

The Maze2D environments test long-horizon planning with sparse rewards. Diffuser achieves significantly higher scores than prior model-free algorithms and even MPPI (which uses ground-truth dynamics). This highlights Diffuser's ability to overcome the difficulties of credit assignment in sparse reward settings, a known weakness of shooting-based approaches.

For the Multi2D setting (randomized goal locations), Diffuser maintains its high performance without needing retraining, simply by changing the conditioning goal. This demonstrates its inherent multi-task planning capability. In contrast, even the best model-free baseline (IQL adapted with hindsight experience relabeling) shows a substantial performance drop in the multi-task setting. The poor performance of MPPI (using ground-truth dynamics) underscores the inherent difficulty of long-horizon planning, even without model inaccuracies, further emphasizing Diffuser's learned planning advantage.

The following are the results from Table 1 of the original paper:

Analysis:

• Diffuser consistently achieves scores well above 100 in all Maze2D and Multi2D settings, indicating it outperforms even the expert policy (where 100 usually denotes expert performance). The high scores (e.g., 119.5 for U-Maze, 129.4 for Medium) suggest Diffuser finds more optimal paths or fewer failures than the reference expert.
• In Maze2D single-task, Diffuser (average 119.5) significantly outperforms IQL (47.0), MPPI (16.2), and CQL (7.7).
• In Multi2D multi-task, Diffuser's performance remains strong (average 116.1), while IQL (16.9) and MPPI (21.5) show a considerable drop compared to Diffuser. CQL results are not provided for Multi2D.
• The comparison with MPPI (which uses ground-truth dynamics) is particularly insightful. Diffuser's superior performance, despite learning a model from data, suggests that its integrated planning mechanism is better suited for long-horizon problems than even a perfect dynamics model coupled with a traditional optimizer in these challenging sparse-reward settings.

6.1.2. Test-time Flexibility (Block Stacking)

In block stacking tasks, which demand adaptation to new goal configurations, Diffuser substantially outperforms both BCQ and CQL. The conditional settings (Conditional Stacking and Rearrangement), which explicitly require flexible behavior generation beyond what was seen in training, prove especially difficult for the model-free algorithms, while Diffuser maintains strong performance by modifying its perturbation function.

The following are the results from Table 3 of the original paper:

Analysis:

• Diffuser demonstrates strong performance across all block stacking tasks, with average scores around 50-60, indicating successful execution of a significant portion of the tasks. The standard error shows reasonable stability.
• BCQ performs at 0.0 in all tasks, suggesting it fails entirely, likely due to the need for out-of-distribution generalization or flexible planning which behavioral cloning struggles with.
• CQL shows some performance in Unconditional Stacking (24.4) but completely fails (0.0) in the more complex Conditional Stacking and Rearrangement tasks that require goal-conditioned or flexible planning.
• This clear differentiation highlights Diffuser's superior ability to generalize and adapt to novel test-time objectives by leveraging its flexible perturbation functions (defined in Appendix B), which are not present in the model-free baselines.

6.1.3. Offline Reinforcement Learning (D4RL Locomotion)

In the D4RL locomotion suite, Diffuser performs comparably to prior algorithms in the single-task setting. It outperforms model-based methods like MORE and MBOP, and return-conditioning methods like DT. However, it performs slightly worse than the best model-free techniques specifically designed for single-task performance (e.g., CQL, IQL, TT in some cases). A crucial finding is that using Diffuser solely as a dynamics model in conventional trajectory optimizers (like MPPI) yielded random performance, reinforcing the conclusion that Diffuser's effectiveness comes from its coupled modeling and planning, not just improved open-loop predictive accuracy.

The following are the results from Table 2 of the original paper:

Analysis:

• In the Medium-Expert datasets, Diffuser achieves competitive scores, particularly in Hopper (103.3) and Walker2d (106.9), where it performs very close to or surpasses top baselines like CQL, IQL, and DT. It's slightly lower than TT and MBOP in HalfCheetah.
• In Medium datasets, Diffuser also performs well, especially in Hopper (74.3) and Walker2d (79.6), where it often surpasses CQL, DT, and TT.
• In Medium-Replay datasets, Diffuser is strong in Hopper (93.6) and Walker2d (70.6), but shows lower performance in HalfCheetah (37.7) compared to some baselines like CQL and TT.
• Overall, Diffuser achieves an average score of 77.5, which is highly competitive, surpassing BC, MOPO, MORel., and MBOP. It is on par with CQL (77.6) and IQL (77.0) and slightly below TT (78.9).
• The results reinforce that Diffuser is a robust offline RL agent, capable of extracting effective policies from diverse datasets. Its performance being comparable to specialized offline RL algorithms designed for single-task performance is a strong indicator of its general applicability. The finding that Diffuser as a mere dynamics model with MPPI fails entirely is critical: it proves that the integration of modeling and planning is the source of its strength, not just the quality of its underlying dynamics prediction.

6.2. Ablation Studies / Parameter Analysis

Warm-Starting Diffusion for Faster Planning

The paper acknowledges a limitation of Diffuser: individual plans can be slow to generate due to the iterative denoising process. To address this, an ablation study is conducted on warm-starting the planning process.

• Method: Instead of always starting from pure noise, a previously generated plan is partially noised and then denosied for a limited number of steps to regenerate an updated plan. This reuses information from the previous timestep's plan.

• Results: The study investigates the trade-off between performance and runtime budget by varying the number of denoising steps (from 2 to 100) when warm-starting. It finds that the planning budget can be significantly reduced with only a modest drop in performance.

  The following figure (Figure 7 from the original paper) illustrates the results of this ablation study:

  

  Analysis:

• The plot shows that even with a significantly reduced number of diffusion steps (e.g., 10 or 20 steps, which is one-tenth or one-fifth of the typical 100 steps), the performance (normalized score) remains high, dropping only slightly from the maximum.

• For instance, reducing steps from 100 to 10 or 20 maintains a score close to 100, while dramatically decreasing the planning budget (runtime).

• This suggests that warm-starting is an effective strategy for speeding up Diffuser's execution in a receding-horizon control loop, making it more practical for real-time applications where planning latency is critical. The iterative nature of diffusion, combined with the temporal coherence of plans, allows for efficient "refinement" rather than full regeneration at each step.

6.3. Implementation Details

Appendix C provides critical hyperparameters and architectural details:

• Architecture: U-Net with 6 residual blocks, each containing two temporal convolutions, group normalization, and Mish nonlinearities. Timestep embeddings are added via a fully-connected layer.

• Training: Adam optimizer (Kingma & Ba, 2015) with learning rate 4e-05 and batch size 32 for 500k steps.

• Reward Predictor ($\mathcal{J}$): First half of the U-Net structure from Diffuser, followed by a linear layer for scalar output.

• Planning Horizon ($T$): Variable across tasks: 32 for locomotion, 128 for block-stacking and Maze2D U-Maze, 265 for Maze2D Medium, and 384 for Maze2D Large.

• Diffusion Steps ($N$): 20 for locomotion, 100 for block-stacking.

• Guide Scale ($\alpha$): 0.1 for most tasks, 0.0001 for hopper-medium-expert. A smaller guide scale for shorter horizons (e.g., 0.001 for horizon 4 in HalfCheetah).

• Discount Factor: 0.997 for return prediction, but performance is robust to changes above 0.99.

• Noise vs. Data Prediction: Performance was not substantially affected by whether the model predicts noise ($\epsilon$) or uncorrupted data ($\tau^0$).

  These details highlight careful tuning and architectural choices that contribute to Diffuser's robust performance.

7. Conclusion & Reflections

7.1. Conclusion Summary

This paper introduces Diffuser, a novel denoising diffusion probabilistic model designed specifically for trajectory data. The core innovation lies in unifying the process of modeling and planning, such that sampling from Diffuser (guided by auxiliary perturbation functions) inherently performs trajectory optimization. This framework reinterprets concepts like classifier-guided sampling for reward maximization and image inpainting for constraint satisfaction as coherent planning strategies. The study highlights several valuable properties of Diffuser, including its ability to handle sparse rewards and long-horizon decision-making, offer test-time flexibility by planning for new reward functions without retraining, and exhibit temporal compositionality for generalizing to novel trajectories by combining familiar subsequences. Empirical evaluations demonstrate Diffuser's superior performance in Maze2D environments (long-horizon, multi-task planning) and block stacking tasks (test-time flexibility), and competitive performance in D4RL offline locomotion benchmarks, pointing towards a promising new class of diffusion-based planning procedures for deep model-based reinforcement learning.

7.2. Limitations & Future Work

The paper implicitly points to one key limitation through its investigation into warm-starting:

• Planning Speed: Generating individual plans with Diffuser can be slow due to the iterative denoising process. While warm-starting helps mitigate this by reusing previous plans and reducing the number of denoising steps, the inherent iterative nature still poses a challenge for real-time applications requiring very low latency.

  The paper suggests the potential for future work by establishing a new paradigm:

• New Class of Planning Procedures: The success of Diffuser opens up a "new class of diffusion-based planning procedures for deep model-based reinforcement learning." This implies further research into optimizing these models, exploring alternative guidance mechanisms, or applying them to a broader range of complex control problems.

7.3. Personal Insights & Critique

This paper presents a highly innovative and refreshing perspective on model-based reinforcement learning. The idea of deeply integrating the planning process into the generative model itself, rather than treating them as separate components, is a significant conceptual leap. It effectively addresses the long-standing problem of learned models being exploited by traditional optimizers, by designing a model whose very sampling mechanism is robustly optimized for planning.

One key insight is the elegant reinterpretation of classifier guidance and inpainting—techniques from generative image modeling—as powerful strategies for reward maximization and constraint satisfaction in sequential decision-making. This cross-pollination of ideas from different subfields of AI is particularly compelling. The temporal compositionality property, allowing the model to stitch together learned behaviors, is also very powerful for generalization, reminiscent of how intelligent agents might recombine motor primitives.

Potential areas for improvement or further investigation:

1. Computational Cost of Guidance: While guided sampling is powerful, calculating the gradients of the reward function $\nabla \mathcal{J}(\mu)$ at each diffusion step can be computationally expensive, especially for complex reward functions or very long horizons. Exploring more efficient guidance mechanisms or approximation techniques could be valuable.

2. Reward Function Learning Robustness: The effectiveness of guided sampling relies on an accurate reward predictor $\mathcal{J}_{\phi}$. If this predictor is noisy or inaccurate, it could lead the diffusion process astray. Research into robust reward learning or uncertainty-aware guidance might be beneficial.

3. Real-world Deployment Challenges: The warm-starting study addresses runtime, but deploying Diffuser in highly dynamic, real-world robotic systems might still face challenges related to the iterative nature of denoising, especially if the environment changes drastically within a single planning cycle. Further work on real-time guarantees or adaptive planning horizons could be explored.

4. Theoretical Guarantees: While empirically strong, further theoretical analysis of the convergence properties and optimality guarantees of diffusion-based planning, especially under various forms of guidance, would strengthen the foundation of this new class of methods.

5. Offline Data Quality Sensitivity: Although tested on D4RL with heterogeneous data, a deeper dive into Diffuser's sensitivity to extremely poor quality or sparse offline datasets would be insightful. The model might perform exceptionally well on expert-generated trajectories but struggle where data quality is severely limited or lacks specific behaviors.

   Overall, Diffuser represents a major step forward, demonstrating that generative models can be more than just dynamics predictors; they can be intelligent planners themselves, opening exciting avenues for flexible and robust behavior synthesis.