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DIME:Diffusion-Based Maximum Entropy Reinforcement Learning

Onur Celik, Zechu Li, Denis Blessing, Ge Li, Daniel Palenicek, Jan Peters, Georgia Chalvatzaki, Gerhard Neumann

arXiv:2502.02316v2cs.LG

TL;DR

MaxEnt-RL benefits from exploration, but Gaussian policies limit representational capacity and diffusion policies make marginal entropy difficult to compute. DIME derives a lower bound on the MaxEnt objective using approximate inference and proposes convergent diffusion-policy iteration, outperforming diffusion baselines across 13 high-dimensional control benchmarks while remaining competitive with Gaussian-policy RL methods.

  • Problem

    Gaussian policy parameterizations significantly limit representational capacity, while diffusion policies pose challenges for MaxEnt-RL because their marginal entropy is difficult to compute.

  • Method

    DIME uses approximate inference with diffusion models to derive a lower bound on the MaxEnt objective and restricts policy iteration to diffusion policies.

  • Results

    DIME significantly outperforms other diffusion-based baselines on all 13 challenging continuous high-dimensional control environments and outperforms Gaussian-policy state-of-the-art RL methods on 10 of 13.

  • Takeaways & Limitations

    DIME provides an expressive diffusion-policy approach to MaxEnt-RL with computational efficiency and fewer algorithmic design choices than BRO.

  • Takeaways & Limitations

    The theoretical analysis assumes finite action spaces, although the paper relaxes this assumption in Section 4.3.

Abstract

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Maximum entropy reinforcement learning (MaxEnt-RL) has become the standard approach to RL due to its beneficial exploration properties. Traditionally, policies are parameterized using Gaussian distributions, which significantly limits their representational capacity. Diffusion-based policies offer a more expressive alternative, yet integrating them into MaxEnt-RL poses challenges-primarily due to the intractability of computing their marginal entropy. To overcome this, we propose Diffusion-Based Maximum Entropy RL (DIME). \emph{DIME} leverages recent advances in approximate inference with diffusion models to derive a lower bound on the maximum entropy objective. Additionally, we propose a policy iteration scheme that provably converges to the optimal diffusion policy. Our method enables the use of expressive diffusion-based policies while retaining the principled exploration benefits of MaxEnt-RL, significantly outperforming other diffusion-based methods on challenging high-dimensional control benchmarks. It is also competitive with state-of-the-art non-diffusion based RL methods while requiring fewer algorithmic design choices and smaller update-to-data ratios, reducing computational complexity.

1. Introduction

DIME addresses the tension between MaxEnt-RL’s exploration benefits and the limited expressiveness of Gaussian policies by using diffusion-based policies despite their intractable marginal entropy. It derives a tractable lower-bound-based approach and achieves strong performance across high-dimensional control benchmarks.

  • MaxEnt-RL augments task rewards with policy entropy, providing improved exploration and supporting sample-efficient, high-performing model-free algorithms.
  • Gaussian policy parameterizations limit representational capacity, whereas diffusion models can represent complex behavior policies more expressively.
  • Diffusion policies are difficult to integrate into RL because their marginal entropy is intractable to compute.
  • DIME derives a lower bound on the MaxEnt objective using approximate inference with diffusion models and introduces a policy iteration framework with monotonic improvement that converges to the optimal diffusion policy.

2. Related Work

Prior work applies diffusion models to offline and online RL, but often relies on behavior cloning, score matching, or added noise for exploration. DIME instead combines diffusion policies directly with the maximum entropy RL framework to control exploration without those additional approximations.

  • Maximum Entropy RL: Maximum entropy RL adds policy entropy at each time step to induce principled exploration, distinguishing it from entropy-regularized RL’s current-step objective.
  • Maximum Entropy RL: Existing maximum entropy approaches include Soft-Q Learning, SAC, SVGD-based energy policies, and Langevin-based exploration, with SVGD approaches harder to scale to high-dimensional control.
  • Diffusion-Based Policies in RL: Diffusion models have been used in offline RL as trajectory generators and expressive policy representations, while online methods include behavior-cloning-based DIPO and score-based QSM.
  • Diffusion-Based Policies in RL: DIPO relies on diffusion stochasticity for exploration without explicitly controlling it through an objective, whereas QSM disregards policy entropy.
  • Diffusion-Based Policies in RL: DIME is the first described approach here to train diffusion policies within MaxEnt-RL, directly controlling exploration-exploitation and generating non-Gaussian exploration actions.
  • Approximate Inference with Diffusion Models: The formulation can use different diffusion samplers, but this work restricts experiments to the sampler presented by Berner et al.

3. Preliminaries

MaxEnt-RL combines rewards with policy entropy to promote exploration, while diffusion policies provide greater representational capacity but require approximate inference for tractable training. The preliminaries describe policy iteration, diffusion noising and denoising, and the difficulty of applying score matching when target samples are unavailable.

  • Maximum Entropy Reinforcement Learning: MaxEnt-RL augments per-step rewards with policy entropy, with α controlling the exploration–exploitation trade-off.
  • Maximum Entropy Reinforcement Learning: Policy iteration alternates evaluation and improvement; repeated Bellman backups converge to Q^π, while the improvement step yields a policy with nondecreasing Q-values and converges to the optimal policy.
  • Denoising Diffusion Policies: Diffusion policies use a forward Ornstein–Uhlenbeck noising process that approaches N(0, η^2I), then a time-reversed process to generate samples from the target policy.
  • Denoising Diffusion Policies: The diffusion time variable t is distinct from the reinforcement-learning time step, with the distinction becoming explicit after discretization.
  • Denoising Diffusion Policies: The reward-scaling parameter α directly controls exploration: larger α smooths the target distribution and produces noisier late-stage samples, whereas smaller α yields more concentrated targets.
  • Denoising Diffusion Policies: Diffusion score matching is ordinarily trained with computable conditional scores, but MaxEnt-RL target samples proportional to exp Q^π are unavailable, so DIME instead relies on Qπ evaluations.

4. Diffusion-Based Maximum Entropy RL

DIME casts maximum-entropy learning for diffusion policies as approximate inference, using an entropy lower bound to address intractable marginal entropy. Its diffusion-policy iteration alternates evaluation and improvement with provable convergence, while the practical algorithm uses function approximation, replay data, and automatic temperature tuning.

  • DIME expresses the maximum-entropy objective as an approximate-inference problem and develops a practical algorithm for optimizing diffusion models.
  • 4.1. Control as Inference for Diffusion Policies: An entropy lower bound replaces the intractable marginal entropy of the denoising diffusion policy.The bound is derived after discretizing the noising and denoising processes with Euler–Maruyama transitions.
  • 4.1. Control as Inference for Diffusion Policies: The target diffusion policy has a marginal proportional to the exponentiated Q-function, connecting approximate inference to maximum-entropy policy optimization.
  • 4.2. Diffusion-based Policy Iteration: Alternating evaluation and improvement provably converges to the optimal policy within the restricted diffusion-policy family.The theoretical analysis assumes finite action spaces, while continuous-control practicality is addressed separately.
  • 4.2. Diffusion-based Policy Iteration: Policy evaluation converges under repeated Bellman backups, and policy improvement produces a policy with no lower Q-value at every state-action pair.
  • 4.3–4.4. Practical Algorithm and Implementation Details: The practical method parameterizes the Q-function and policy, trains from replay-buffer state-action pairs, and optimizes the policy without samples from the exponentiated Q-function.Automatic temperature tuning adjusts the entropy-scaling parameter α using a target mismatch between noising and denoising processes.

5. Experiments

Experiments evaluate DIME across 13 challenging locomotion and manipulation environments using diffusion and Gaussian baselines. DIME benefits from tuned exploration, sufficient diffusion steps, and expressive policies, achieving strong results across Gym, DMC, and MYO benchmarks with lower training cost than BRO.

  • 5. Performance Evaluation: Experiments cover 13 challenging environments from MuJoCo Gym, DMC, and MYO, with action spaces up to 39 dimensions and observation spaces up to 223 dimensions.Results use 10 seeds and report the interquartile mean with a 95% stratified bootstrap confidence interval.
  • 5.1. Ablation Studies: α = 0.1 prevents learning, α ≤10−5 yields suboptimal behavior, and performance peaks at α = 10−3 on DMC dog-run.
  • 5.1. Ablation Studies: The diffusion policy achieves higher aggregated return on humanoid-run and significantly faster convergence on high-dimensional dog-run than an identically implemented Gaussian policy.The comparison changes only the policy representation and corresponding policy update.
  • 5.1. Ablation Studies: 16 and 32 diffusion steps perform similarly and better than fewer settings, while increasing diffusion steps increases runtime until performance saturates from 16 steps onward.
  • 5.2. Performance Comparisons: On Ant-v3, DIME performs comparably to CrossQ and outperforms diffusion baselines; on Humanoid-v3, DIME achieves significantly higher return than all baselines.
  • 5.2. Performance Comparisons: DIME significantly outperforms all baselines on DMC dog-run, converges faster on other dog tasks, and performs competitively across humanoid and MYO environments.BRO averages 8.5h of training versus approximately 4.5h for DIME on humanoid-run with 16 diffusion steps and the same hardware.

6. Conclusion and Future Work

DIME applies approximate inference with diffusion models to maximum entropy reinforcement learning and presents future directions based on alternative diffusion processes and inference methods.

  • The work uses denoising diffusion models whose forward process follows an Ornstein-Uhlenbeck process.
  • Future Work: Future directions include learning both diffusion processes, incorporating target-density gradients, and combining diffusion models with Sequential Monte Carlo.

A. Derivations

The derivation reformulates the maximum entropy objective using diffusion-policy inference relationships and a nonnegative KL-divergence bound.

  • Approximate Inference Formulation: A relation between diffusion-policy terms is used to rewrite the objective in an approximate-inference formulation.
  • The lower-bound reasoning relies on the fact that KL divergence is always non-negative.
  • The derivation begins by recalling the Q-function and reformulating the maximum entropy objective.

B. Proofs

The proofs establish policy evaluation and improvement properties for diffusion policies, yielding monotonic convergence to the optimal maximum entropy policy.

  • Policy Evaluation: Policy evaluation uses an entropy-augmented reward and a Q-function update rule.
  • Policy Evaluation: Standard convergence results apply to the policy-evaluation formulation.
  • Policy Improvement: The policy-improvement proof uses nonnegative KL divergence to establish the improvement property.
  • The notation is simplified during the derivation to keep expressions uncluttered.
  • Policy Iteration: Repeated policy evaluation reaches a fixed point Q^(i+1) = Q^(i) = Q*.
  • Policy Iteration: Policy iteration converges to the optimal policy because Q for the updated policy is at least as large as for the previous policy.

C. Environments

The evaluation covers locomotion and manipulation environments from the MuJoCo Gym, DeepMind Control Suite, and Myo Suite benchmarks.

  • The environments include Ant-v3 and Humanoid-v3 from MuJoCo Gym, seven locomotion tasks from DMC, and four hand tasks from Myo Suite.The DMC tasks are humanoid-stand, humanoid-walk, humanoid-run, dog-stand, dog-walk, dog-trot, and dog-run; the Myo tasks are object-hold-random, reach-random, key-turn-random, and pen-twirl-random.

D. Implementation Details

DIME’s implementation uses score-based diffusion policies, distributional Q learning, bounded-action squashing, and an update procedure with delayed policy updates. Hyperparameters and baseline settings are specified across benchmark suites.

  • Policy and diffusion: DIME uses a 3-layer score network with 256-dimensional hidden layers, GELU activations, Fourier timestep features, and a cosine diffusion schedule.The diffusion coefficient scaling parameter β is optimized per action dimension end-to-end.
  • Critic: Distributional Q learning represents the Q-function with probabilities over bins and updates it using entropy-regularized cross-entropy.A small regularization improves early learning but does not change asymptotic performance.
  • Action scaling: The unbounded diffusion action is transformed with tanh at the final diffusion step so actions lie in (−1, 1), with a corresponding likelihood correction.The Gaussian kernels retain their log probabilities except for the final-step correction.
  • Optimization: Algorithm 1 updates the critic and applies delayed policy and temperature updates, with UTD denoting the update-to-data ratio.Policy delay is the number of policy updates delayed relative to critic updates.
  • Benchmark configuration: The implementation uses benchmark-specific distributional-Q bin ranges and reports separate hyperparameter tables for diffusion-based and Gaussian-based algorithms.Baseline-specific adjustments include QSM tuning and modified CrossQ settings for some benchmark suites.

F. General Diffusion Policies

DIME’s maximum-entropy framework extends beyond denoising diffusion to general diffusion policies by parameterizing forward and backward processes with control functions. Preliminary experiments apply the same optimization framework to these controls.

  • General diffusion policies: DIME can extend its maximum-entropy framework from denoising diffusion policies to general diffusion policies through the General Bridges framework.The extension is formulated using forward and backward processes.
  • Process formulation: The forward process uses drift f, control u, diffusion coefficient β, and Brownian noise, while the backward process starts from a standard normal distribution.The forward and conditional marginal densities are denoted π_t and π_t|l.
  • Discretization: Euler–Maruyama discretization updates the state using the drift and forward control, plus Gaussian noise with covariance 2βδI.The discretization uses step size δ with N = T/δ.
  • Transition kernels: The discretized forward and backward transitions are Gaussian kernels whose means incorporate the respective control functions u and v.Their conditional distributions are given explicitly for adjacent diffusion steps.
  • Optimization and results: The controls u and v are optimized with the same objective framework used in the main method, with the target policy specified at n = 0.In practice, both controls are represented by parameterized neural networks.
  • Optimization and results: Preliminary general-bridge results are reported for the DMC dog-run and humanoid-run environments, including a comparison with BRO on humanoid-run for 3 million steps.The accompanying figures visualize learning curves and learned β parameters during training.

G. Additional Experiments

Additional experiments test DIME with and without distributional Q learning against diffusion and Gaussian-policy baselines. The results show broadly favorable performance, with distributional Q providing only a small improvement in the reported dog tasks.

  • Distributional Q comparisons: DIME converges faster than DACER to the same Ant-v3 performance and outperforms diffusion-based baselines on Humanoid-v3 when using distributional Q learning.The comparisons include distributional variants of Diff-QL and Consistency-AC.
  • Distributional Q comparisons: Without distributional Q learning, DIME performs similarly to DIPO and QVPO on Ant-v3 and outperforms all baselines on the higher-dimensional Humanoid-v3 task.This setting updates the Q-function with the residual Bellman function.
  • DMC dog environments: Distributional Q produces a small improvement for DIME on the four DMC dog environments.The experiment compares DIME with and without distributional Q against BRO and CrossQ.
  • DMC dog environments: DIME without distributional Q performs on par with or better than BRO and consistently outperforms BRO (Fast) and CrossQ.BRO and BRO (Fast) use quantile distributional reinforcement learning, whereas CrossQ uses Bellman residual loss.
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