Symmetry-Aware Steering of Equivariant Diffusion Policies: Benefits and Limits

Reinforcement learning (RL) post-training provides a direct way to align diffusion models with human preferences and task-specific rewards. However, current RL algorithms for diffusion models remain fragmented: reverse-trajectory methods rely on discretized likelihood ratios, whereas forward-matching methods train on reward-labeled noising versions of the rollout samples. This paper shows that these seemingly different losses arise from a single path-space principle. Starting from the regularized diffusion-RL objective, we use importance sampling between sampling SDEs to obtain an explicit policy-gradient estimator on trajectory space. The estimator contains the stochastic Itô integral underlying Flow-GRPO-type updates; we derive an equivalent variance-reduced value-gradient form that recovers the forward-matching structure of AWM and DiffusionNFT. This identifies the empirical gap between these method families as a variance-reduction effect rather than a difference in RL principle. The derivation yields a unified design space organized by value-gradient estimation, weight functions, and sampling choices. Within this space, we propose a multi-sample KDE value-gradient estimator that reuses rollout groups, together with scale-bounded weight families that retain stable existing recipes while excluding singular ones. Experiments on SD3.5-M and Qwen-Image models validate the variance-reduction explanation and show that the resulting recipe improves over prior diffusion-RL baselines.

Step distillation accelerates diffusion sampling by training a few-step student to imitate a many-step teacher, but distillation itself remains expensive. Typically, this requires thousands of GPU-hours and a large pre-generated trajectory dataset. We introduce ReDiF, which casts step distillation as terminal-reward policy optimization rather than step-wise regression. The student is optimized against a reward computed on the terminal sample, measuring alignment with the teacher's output, instead of matching the teacher's intermediate trajectory under a reconstruction or consistency loss. Because the reward need not be differentiable or trajectory-aligned, ReDiF admits non-differentiable objectives, multi-objective combinations, and preferences the teacher does not express, while exploration lets the student find sampling paths matched to its own step schedule. ReDiF converges in 400 policy updates with 3200 rollouts on a single A100 GPU with 1,000 noise-class pairs and no paired dataset: about 4 GPU-hours, against roughly 336 A100-hours reported for DMD2 on the same EDM teacher. At 8 steps on ImageNet-64, it achieves an FID 3.64 points better than the strongest retrained distillation baseline in the low-training regime under the same single-GPU budget. The formulation is also orthogonal to existing distillation objectives: added to the DMD2 loss, it further improves DMD2's FID by 4.4.

We propose Parameterized Diffusion Policy (PDP), a framework for learning diffusion policies conditioned on low-dimensional, continuous parameters embedded in a learned behavior manifold. By constructing this manifold so that distances between latent representations reflect the semantic similarity between physical trajectories, we transform diffusion from a mechanism for stochastic diversity into a precise and optimizable tool for behavior steering. Our approach enables smooth interpolation between known strategies and efficient adaptation to novel constraints without updating policy weights. We demonstrate that PDP significantly improves adaptation performance on complex multimodal benchmarks in both simulated and real-robot experiments compared to standard diffusion policies, particularly in scenarios requiring the synthesis of novel behaviors.