Equivariant Action Sampling for Reinforcement Learning and Planning

Jung Yeon Park; Lawson L.S. Wong; Linfeng Zhao; Owen Howell; Robin Walters; Xupeng Zhu; Zhewen Zhang

REVIEW 4 major objections 5 minor 69 references

Equivariant Action Sampling for Reinforcement Learning and Planning

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper shows that expanding each sampled action into its group orbit makes sampling-based action selection exactly equivariant for any finite sample count, and that an equivariant model-based planner built on this operation learns…

desk verdict G-augmented sampling is a clean, correct mechanism for exact finite-sample equivariance, but the paper overclaims on 'always better' and leaves the finite-subgroup implementation gap unquantified. read the letter →

arxiv 2412.12237 v1 pith:TDG7D3E2 submitted 2024-12-16 cs.RO cs.AIcs.LG

Linfeng Zhao , Owen Howell , Xupeng Zhu , Jung Yeon Park , Zhewen Zhang , Robin Walters , Lawson L.S. Wong This is my paper · ORCID

classification cs.ROcs.AIcs.LG

keywords symmetryequivarianceactionsamplingmodel-basedreinforcementlearningmodelpredictivepathintegralcontinuouscontrolgeometricMDPG-augmented

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Sampling-based action selection in continuous control breaks the exact symmetries of the task whenever the sample set is finite; for example, rotating the input state does not reliably rotate the selected action unless the sampler is exhaustively symmetric. This paper introduces $G$-augmented sampling: after drawing actions from any distribution, each action is expanded into its group orbit $\{g\cdot a : g\in G\}$, and the best action is chosen from that orbit. Provided the scoring function (energy, Q-value, or return) is $G$-invariant, the chosen action is exactly $G$-equivariant for any number of samples, not just in the infinite-sample limit. The same construction is lifted to multi-step MPPI planning inside an equivariant TD-MPC, and the resulting algorithm reaches near-optimal reward roughly 2 to 3 times faster, in interaction steps, than the non-equivariant baseline on several goal-reaching tasks. A sympathetic reader should care because this closes the gap between equivariant network design and the stochastic optimizers that actually select actions in model-based reinforcement learning.

What carries the argument

The load-bearing object is the $G$-augmented sample set (the $G$-sample): for each drawn action $a$, the sampler also evaluates every transformed action $g\cdot a$ with $g$ in the symmetry group $G$, so the candidate set is $GA = \{g\cdot a_i : i=1,\dots,N,\ g\in G\}$. Because the scoring function $E(s,a)$ is $G$-invariant, evaluating $\arg\min$ over this orbit-expanded set makes the selection commute with the group action; a rotated state picks the rotated winner. Around this, the paper assembles a geometric MDP formalism: transition dynamics must be $G$-equivariant, reward and value functions $G$-invariant, the policy $G$-equivariant, and the MPPI return functional $G$-invariant, which together make the whole planning procedure equivariant. The other mechanism is the weak/strong equivariance distinction, which diagnoses why finite-sample averages fail and what operation restores the symmetry exactly.

What would settle it

Take a task with $SO(2)$ or $O(2)$ symmetry, implement $G$-augmented sampling with $D_8$ as in the paper, and measure equivariance error for state rotations by angles that are not multiples of $45^\circ$ (e.g., $15^\circ$) using a small fixed sample count; if the selected action fails to rotate by that angle, or the equivariance error jumps above machine precision, the exact-equivariance claim holds only for the discrete subgroup, not for the task's true continuous symmetry.

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Extended reading notes

Core claim

The central claim is that the stochasticity of sampling, not the learned model, is what breaks symmetry in sampling-based control, and that this can be repaired exactly. The paper defines weak equivariance (holding only after averaging over samples) versus strong equivariance (holding for every finite sample set), and shows that a weakly equivariant estimator can be converted into a strongly equivariant one by symmetrization. Concretely, for action selection $a_0 = \arg\min_{a\in A} E(s,a)$, replacing the sample set $A$ with the $G$-augmented set $GA = \{g\cdot a : a\in A,\ g\in G\}$ makes the map $s\mapsto a_0$ $G$-equivariant whenever $E$ is $G$-invariant: rotating the state rotates the selected action, for any sample budget. The paper extends this to trajectory planning by requiring $G$-equivariant dynamics, $G$-invariant reward and value networks, a $G$-equivariant policy, and $G$-augmented sampling inside MPPI; Proposition 1 states that the return is $G$-invariant and the $G$-sample procedure is $G$-equivariant. Empirically, an equivariant TD-MPC built from these ingredients matches or beats the non-equivariant TD-MPC and reaches near-optimal reward 2 to 3 times faster (in interaction steps) on PointMass, Reacher, 3D PointMass variants, and MetaWorld Reach. A further claim, proved in the appendix, is that a symmetrized policy or estimator is never worse, in $G$-averaged error, than the unsymmetrized one.

Load-bearing premise

The load-bearing premise is the paper's own stated choice (Section 6.1) to replace each task's true continuous symmetry group with a finite discrete subgroup ($D_4$, $D_8$, $C_8$, icosahedral, or octahedral); the exact equivariance guarantee and the empirical gains are proven only for those discrete group elements, and the derivations in Sections 4.4 and 5.3 assume the full group $G$ with no error bound for the discretization.

Editorial extensions

If this is right

With a $G$-invariant score, $G$-augmented sampling gives exact equivariance for any finite sample count, including a single sample, so the usual infinite-sample asymptotic is not needed.
An equivariant TD-MPC instantiated with discrete subgroups ($D_4$, $D_8$, $C_8$, icosahedral, octahedral) learns near-optimal goal-reaching policies 2 to 3 times faster in interaction steps than non-equivariant TD-MPC on the tested tasks.
In coordinate regression, combining a $G$-invariant energy with $G$-augmented sampling enables spatial extrapolation beyond the training region's convex hull, where the non-equivariant baseline fails.
The symmetrized estimator is provably no worse than the raw estimator in $G$-averaged error, so enforcing symmetry in the sampler does not sacrifice expected accuracy.
Larger discrete subgroups (e.g., $D_{16}$ vs. $D_8$) plateau in performance while increasing cost, suggesting the chosen subgroups already capture the task-relevant symmetry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The $G$-augmentation recipe is generic: any sampling-based optimizer that scores candidates by an invariant function can be made strongly equivariant by orbit expansion, so the result should transfer to planners beyond the TD-MPC instantiation the paper tests.
A direct test of the finite-subgroup assumption would measure equivariance error for rotations not in the implemented subgroup (e.g., $15^\circ$ for $D_8$); the paper's guarantees only cover the discrete group elements, so intermediate angles may reveal how much the surrogate matters.
Because symmetrization multiplies the candidate set by $|G|$, the computational cost grows with group size; on large-symmetry problems one might trade exactness for cost by subsampling the orbit, a trade-off the paper does not analyze.
The paper's own ablation shows that on kinematically constrained tasks with global reference frames, equivariance can hurt; the practical benefit likely depends on representing features in the right local frame rather than on equivariance alone.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the core idea is real and worth engaging with. Expanding each sampled action to its group orbit and then selecting the arg-min/max makes action selection exactly equivariant for any finite sample size, provided the score is invariant. That is simple, correct, and demonstrated cleanly in the coordinate regression experiment (Fig. 3). The weak/strong equivariance distinction is a useful framing, and applying orbit averaging to finite-sample action selection in continuous MDPs is a fair novelty, even if the underlying math is classical. The equivariant TD-MPC results show consistent 2-3x sample-efficiency gains on symmetric control tasks, and the appendix ablations (equivariant Q, policy, transition, planning-free, group choice) are more thorough than typical. The authors also honestly note in B.2 that kinematic constraints can limit the benefit.

Soft spots, in proportion. First, the 'always better' claim in D.2 is overstated. The proof shows that the group-averaged error of the symmetrized policy is no larger than the group-averaged error of the original. That is not 'always better' in the usual per-instance sense. The text should say 'better in group-averaged error.' Second, the theory is stated for a compact group G, but the implementation uses finite subgroups (D4, D8, C8, icosahedral, octahedral). The stress-test is right: for h in G\H, the selected action is not guaranteed to transform correctly, and no error bound is given. The paper acknowledges discretization in Section 6.1 but does not analyze it. This is a real gap between theory and implementation, especially since the tasks have continuous symmetry. It is not fatal—the method still gives exact H-equivariance, and D8 evidently captures enough symmetry to produce the gains—but the claims of exact equivariance should be scoped to the implemented subgroup. Third, the planning experiments do not isolate the G-augmented sampling contribution from the equivariant network components. Figure 9 ablates equivariant Q/pi/transition, but not G-sample on/off. A direct ablation of G-augmented sampling alone would clarify the mechanism's contribution. No code is shipped.

This paper is for researchers working on equivariant RL/planning and practitioners with symmetric control tasks. It deserves a serious referee: the core idea is correct, the experiments are mostly well done, and the limitations are at least partly acknowledged. I would recommend conditional acceptance: fix the 'always better' overclaim, scope the theory to the implemented subgroup or add an approximation bound, and consider an ablation isolating G-sample.

Referee Report

4 major / 5 minor

Summary. The paper proposes G-augmented sampling for sampling-based action selection in reinforcement learning and planning. Instead of evaluating only N sampled actions, the method evaluates all group transforms of each sample, i.e., the orbit set GA = {g·a_i}, and then selects the best action from this orbit. The authors show that, for an invariant energy/return function E, this selection rule is exactly G-equivariant for any finite number of samples, provided the argmin is unique or ties are handled equivariantly. They apply the idea to an energy-based coordinate regression task and to an equivariant version of TD-MPC with MPPI planning on several continuous control tasks, reporting that the equivariant version reaches near-optimal performance 2 to 3 times faster in training interactions. The appendix contains proofs of equivariance of the Bellman operator for geometric MDPs, a 'symmetrized policy is always better' argument, and additional ablations.

Significance. The core algebraic observation is elegant and useful: for a fixed finite sample set, enlarging it to the full group orbit makes the argmin-based selection equivariant by construction, with no dependence on the number of samples. If the remaining gaps are closed, the paper would provide a practical recipe for adding symmetry to sampling-based planners such as CEM and MPPI, and the benchmark suite is relevant to the RL and robotics communities. The paper is also honest in reporting ablations on the equivariant components, the choice of finite subgroups, and the warm-up sensitivity. However, the theoretical claims currently outrun the proofs in three load-bearing places: the finite-subgroup implementation, the 'always better' statement, and the treatment of randomness in the sampling procedure.

major comments (4)

[Section 6.1, Proposition 1 (Section 5.3), Eq. (5)] The theory is developed for the full symmetry group G (O(2), SO(2), SO(3)), but Section 6.1 states that all equivariant MLPs are implemented with finite subgroups H in {D4, D8, C8, icosahedral, octahedral}. For any h in G\H, the networks are not exactly equivariant and the orbit set HA is not closed under h, so Eq. (5) and Proposition 1 do not hold for h. The paper provides no bound on the equivariance error as a function of how densely H approximates G, the Lipschitz constant of E, or the sample count. Please either prove an approximation guarantee or explicitly restate the theoretical results as H-equivariance rather than G-equivariance.
[Section 4.3, Appendices D.1 and D.2] The claim in Section 4.3 that the strongly equivariant estimator 'is guaranteed to be a better estimator' and the title of Appendix D.2, 'Equivariant Sampling Is Always Better', are stronger than what the proofs establish. Appendix D.1 proves only the group-averaged inequality ∫_G |H(gx)−Ĥ^G(gx)|dg ≤ ∫_G |H(gx)−Ĥ(gx)|dg, and Appendix D.2 similarly shows that the symmetrized policy is closer on average over group orbits. These results do not imply pointwise improvement for each input x. The main text should either qualify the claim as an improvement in the group-averaged sense or state and prove additional assumptions that yield pointwise dominance.
[Eq. (5), Section 5.3, Appendix E] The proof of Eq. (5) fixes the sample set GA and shows that g·argmin_{a∈GA} E(s,a) equals argmin_{a∈GA} E(g·s,a). In the actual CEM and MPPI procedures described in Sections 4.4 and 5.3, however, the base samples are drawn anew for each state. For a transformed state g·s, the new sample set is only equidistributed with the transformed old set g·GA, not equal to it, so one does not obtain g·a0(s) = a0(g·s) on individual runs unless the random noise itself is transformed as part of an explicitly coupled equivariant map. Please specify the stochastic inputs of the algorithm and prove the exact property that holds; if only distributional (weak) equivariance is achieved, the claim that the procedure 'always preserves equivariance' should be weakened accordingly.
[Eq. (5), Appendix E] The equivariance identity for argmin also assumes a unique minimizer or an equivariant tie-breaking rule. With multiple minimizers, g·argmin and argmin after transformation may select different elements unless tie-breaking is chosen compatibly with the group action. This assumption should be stated explicitly in Proposition 1 and reflected in the algorithm description, since ties are not impossible with finite sample sets and learned energy functions.

minor comments (5)

[Section 1] There is a typo in 'reinforcment learning' in the sentence about equivariant reinforcement learning methods; it should be 'reinforcement learning'.
[Appendix D.3, Eq. (16)-(21)] In step (3) of the Bellman-operator proof, after substituting a = g^{-1}ā, the transition probability still contains g^{-1}a instead of g^{-1}ā; this appears to be a typographical error in an otherwise valid proof sketch.
[Appendix F.2] The paragraph describing the 'sqrt strategy' and 'linear strategy' is garbled: the sentence 'We use two strategies: sqrt strategy ... we use linear strategy ...' is incomplete and the two strategies are described in an overlapping way. Please rewrite this passage for clarity.
[Section 6.2, Figure 6] The claim that equivariant methods achieve near-optimal performance '2 to 3 times faster' is not supported by all panels; for example, the PointMass 2D panel shows only a modest advantage for D8-equivariant TD-MPC. Please quantify the speedup across tasks with confidence intervals or specify which panels support the 2-3x statement.
[Bibliography] Several bibliographic entries contain typos or incomplete metadata, e.g., 'Markov Deision Proesses', 'reinforcment', and the Domshlak entry lacking a year/venue. These should be corrected before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central equivariance statements are derived from explicit symmetry assumptions and validated against external benchmarks; the finite-subgroup implementation is an approximation gap, not a circular argument.

full rationale

The paper's theoretical chain is self-contained. The equivariance of G-augmented sampling (Eq. 5 and Proposition 1) follows directly from the stated assumptions that the energy/Q function is G-invariant and the augmented sample set GA is G-invariant; the proof in Appendix E supplies the algebraic reduction. The 'always better' lemma in Appendix D.2 is a triangle-inequality consequence of the G-averaged policy definition under the cited equivariant-optimal-policy result from van der Pol et al., which is external and not authored by the present authors. Theorem 1 on the equivariant Bellman operator is proved in Appendix D.3 using the GMDP symmetry equations and Haar invariance; the cited prior work by Zhao et al. is contextual, not load-bearing. Empirical claims are made against non-equivariant TD-MPC and naive sampling baselines on standard benchmarks, so no fitted parameter is renamed as a prediction. The main caveat—use of finite subgroups (D4, D8, C8, icosahedral, octahedral) to implement continuous symmetries (Section 6.1)—means the implemented method is exactly H-equivariant rather than exactly G-equivariant, and no error bound is given. That is a correctness/robustness limitation, not circularity, because the implemented algorithm's behavior is not forced by a self-citation or by a parameter fitted to the target result.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The free parameters are modeling choices (discrete symmetry group, capacity scaling, warmup) that the empirical claims depend on. The axioms are standard MDP symmetry assumptions plus the strong implementation assumption that the finite-subgroup equivariance is exact for the chosen representations. The most fragile entry is the discretization axiom, since the theoretical claims are stated for full continuous groups.

free parameters (3)

Symmetry group choice (D4, D8, C8, icosahedral, octahedral) = D8 for 2D tasks, icosahedral for 3D tasks (from ablation in App F.3)
The reported speedups depend on the chosen finite subgroup. The paper ablated C4, C8, C16, and D16 on Reacher Hard and found C8 sufficient, and used D8 in the main results. This is a modeling choice rather than a fitted constant, but it materially affects every empirical result.
Hidden dimension scaling (sqrt strategy) = hidden units divided by sqrt(|G|)
Chosen to keep the number of free parameters roughly equal between equivariant and non-equivariant networks. This capacity normalization affects the comparison and is not derived from theory.
Warmup seed_steps = 1 (no warmup) in the main curves
The paper notes TD-MPC is very sensitive to this hyperparameter and that the equivariant version is robust to it. The main figures use no warmup, which changes the relative performance; sensitivity is explored in App F.3.

assumptions (5)

domain assumption The MDP is a Geometric MDP: transition is G-equivariant and reward is G-invariant (Equations 1 and 2).
This is the foundation of all theoretical results. The paper argues the test tasks have this property, but physical systems only approximately satisfy it (e.g., gripper dynamics, friction, kinematic constraints).
domain assumption The optimal policy satisfies pi*(g·a | g·s) = pi*(a | s), taken from van der Pol et al. 2020b.
Used in Appendix D.2 to show that the symmetrized policy is closer to the optimal policy in a group-averaged metric. The result is cited, not re-derived.
standard math Haar measure on the compact group G is invariant, and the measure on state/action spaces satisfies d(gs) = ds for g in E(d).
Used in the proof of Theorem 1 (step 5) and in the return-invariance derivation in Appendix E. This is standard for Euclidean isometries and compact groups.
ad hoc to paper The learned networks (dynamics, reward, value, policy) are exactly G-equivariant or G-invariant as implemented with the escnn package for the chosen finite subgroup.
The planning procedure's equivariance guarantee depends entirely on the implementation being exact for the finite subgroup; errors in representation matching or in the discretization would break the guarantee. The paper does not provide an error analysis for the discretization.
ad hoc to paper The energy or return function has a unique argmin/argmax, or ties are broken consistently across group transforms.
Equation 5 and Proposition 1 require the argmin/argmax to commute with the group action. With degenerate score functions, ties can break equivariance; the paper does not discuss tie-breaking.

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Cite this review

Pith. "Pith review of Equivariant Action Sampling for Reinforcement Learning and Planning." pith.science (2026). https://pith.science/paper/TDG7D3E2

@misc{pith2026241212237,
  author       = {Pith},
  title        = {Pith review of: Equivariant Action Sampling for Reinforcement Learning and Planning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TDG7D3E2}},
  note         = {Machine review of arXiv:2412.12237}
}

read the original abstract

Reinforcement learning (RL) algorithms for continuous control tasks require accurate sampling-based action selection. Many tasks, such as robotic manipulation, contain inherent problem symmetries. However, correctly incorporating symmetry into sampling-based approaches remains a challenge. This work addresses the challenge of preserving symmetry in sampling-based planning and control, a key component for enhancing decision-making efficiency in RL. We introduce an action sampling approach that enforces the desired symmetry. We apply our proposed method to a coordinate regression problem and show that the symmetry aware sampling method drastically outperforms the naive sampling approach. We furthermore develop a general framework for sampling-based model-based planning with Model Predictive Path Integral (MPPI). We compare our MPPI approach with standard sampling methods on several continuous control tasks. Empirical demonstrations across multiple continuous control environments validate the effectiveness of our approach, showcasing the importance of symmetry preservation in sampling-based action selection.

Figures

Figures reproduced from arXiv: 2412.12237 by the authors.

**Figure 1.** Illustration of the coordinate regression problem (Sec 4.2) and its equivariance. (Left) The energy function EBM takes image and coordinate samples and outputs scalar energy value. (Right) Equivariance in coordinate regression: rotating the image and augmenting samples results in rotated coordinate prediction. et al., 2016]), which is the abstraction of 2D discrete or continuous navigation [PITH_FULL_IMAGE:figures/… view at source ↗

**Figure 2.** Demonstration of results on coordinate regression problem: left two columns for training on entire region, and right three columns for training only on coordinates in first quadrant. action space a ∈ A to select better actions for each state st using methods such as the Cross-Entropy Method (CEM) [de Boer et al., 2005] or Model-Predictive Path Integral [Williams et al., 2017b]. In step (1), the neural network can be… view at source ↗

**Figure 3.** Measuring the equivariance error of using whether G-invariant E(s, a) and whether augment action with G. Analysis: Equivariance Error in 1-Step CEM. To explore the relationship between equivariance and finite samples, we simulate CEM using untrained equivariant and non-equivariant energy functions E(s, a), thereby avoiding any learned equivariance from data. We measure the equivariance error under two conditions: … view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: The proposed sampling-based planning algorithm a0 = plan(s0): if the input state is rotated, the output action should be rotated accordingly. This requires (1) the learned functions to be G-equivariant or G-invariant networks and (2) a specialized sampling strategy, as…

**Figure 5.** Figure 5: Tasks used in experiments: (1) PointMass in 2D, (2) Reacher, (3) Customized 3D version of PointMass with multiple particles to control, and (4) MetaWorld task to reach an object with gripper. 6 Evaluation: Sampling-Based Planning In this section, we present the setup a…

**Figure 6.** Figure 6: (Upper) Results on PointMass, Reacher, and MetaWorld Reach task. (Lower) A set of customized 3D N-ball PointMass tasks, with N = 1, 2, 3, and a customized 3D PointMass with a smaller target. The training procedure follows TD-MPC [Hansen et al., 2022]. We use the state …

**Figure 7.** Figure 7: Commutative Diagram For G-equivariant function: Let Φ(g, ·) : G × Ω → Ω denote the action of G on Ω. Let Φ ′ (g, ·) : G × Ω′ → Ω′ denote the action of G on Ω′ The map Ψ : Ω → Ω′ is G-equivariant if and only if the following diagram is commutative for all g ∈ G [PITH_F…

**Figure 8.** Figure 8: Ablation study on planning component. we aim to build model-based version of equivariant RL algorithm over modelfree counterparts. The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p033_8.png]

**Figure 9.** Figure 9: Ablation study on equivariant components, using [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]

**Figure 10.** Figure 10: Ablation study on number of warmup episodes on [PITH_FULL_IMAGE:figures/full_fig_p035_10.png]

**Figure 11.** Figure 11: Ablation study on symmetry group on Reacher Hard. ory section, we emphasize that kinematic constraints introduce local reference frames. Here, we study a specific example: Reacher (Easy and Hard). The second joint has angle θ2 and angular velocity ˙θ2 relative to the …

**Figure 12.** Figure 12: Results for global reference frame on Reacher. matic constraints (e.g., the length of the second link is preserved in cos θ2,sin θ2), compared to using equivariant feature [PITH_FULL_IMAGE:figures/full_fig_p036_12.png]

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" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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