{"id":"f113a5ff-2c8c-42b3-88e0-c07b024ec405","arxiv_id":"2509.14431","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"LEGO couples agent-centric canonicalization with role-aware graph transformers in MAPPO to produce E(2)-equivariant, zero-shot scalable swarm control policies.","lead":"LEGO is a policy architecture that makes each robot reason in its own local coordinate frame and uses role-based graphs for teamwork, turning MAPPO into an E(2)-equivariant swarm controller. It learns faster and generalizes to new team sizes in cooperative and competitive benchmarks, with a small Crazyflie hardware demo.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. 2's global CoM conflicts with Tag-occlusion partial observability; code must clarify whether c uses full state or visible subset, and an ablation should settle it.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: global CoM in Eq. 2 conflicts with the partial observability of Tag-occlusion. I agree with that read. The paper's formal equivariance argument is built on canonicalizing the full global state X, while the experiment section claims agents see only non-occluded teammates and opponents; the gap between these two descriptions is not resolved anywhere in the manuscript. This is not an internal inconsistency in the mathematics, but an unresolved implementation ambiguity that determines whether the central competitive result is valid under decentralized execution. A code inspection plus a visible-only ablation would settle it. Secondary issues (the zero-velocity fallback in Eq. 1 breaks exact equivariance at a measure-zero state, and prior equivariant MARL baselines are absent) are real but would not change the conditional verdict; the CoM/partial-observability question is the first thing a reader needs answered before the empirical claims can be trusted. Therefore I keep the reader's CONDITIONAL verdict unchanged.","tokens_in":13121,"tokens_out":7692,"duration_ms":71207,"concrete_test":"Inspect the released code at https://github.com/CAB-Lab-Princeton/LEGO-MARL, specifically the Tag-occlusion observation builder, and log at the Eq. 2 computation whether c is built from all N agent positions in the simulator's global state or only from agents present in O_i. If c uses all N positions, rerun the Tag-occlusion experiment with c restricted to the visible subset; a performance drop confirms the reported gains rely on centralized information. If c already uses only visible agents, numerically verify E(2)-invariance by applying random rotations/reflections to the full scene (including obstacles) and checking whether the canonicalized observations and sampled actions are invariant, including at the zero-velocity fallback of Eq. 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section IV-B defines the canonical y-axis using the center of mass c = (1/N) sum_j p_j over all N agents (Eq. 2), and Sections IV-B/E claim the policy is E(2)-equivariant by construction. The Tag-occlusion task (Section V-B) is explicitly partially observable: \"Each agent can only observe teammates and opponents that are not occluded.\" An occluded agent's position is absent from O_i, so the exact Eq. 2 CoM cannot be computed from local observations. The paper never states whether c is formed from the full global state X (Fig. 2 caption canonicalizes \"global observation X\") or from O_i's visible subset. If the full state is used at execution, the actor is not truly decentralized under CTDE, and the competitive results are not a clean partial-observability benchmark. If visible-only positions are used, Eq. 2 is not implemented as written, and the claimed \"by construction\" E(2)-invariance must be re-derived for a varying visible subset; the formal guarantee no longer follows directly. This is load-bearing because the competitive Tag-occlusion result and the paper's \"local canonicalization\" contribution both depend on resolving this ambiguity.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces LEGO, a modular policy architecture for multi-agent reinforcement learning that combines agent-centric canonicalization (to achieve E(2) equivariance), role-based Graphormer encoders (for permutation equivariance and size generalization), and MAPPO as the training algorithm. The authors claim improved sample efficiency and performance over MLP-based, graph-only, and canonicalization-only baselines on the cooperative MPE Spread task and the competitive Tag-occlusion task, zero-shot generalization to unseen team sizes, robustness to spatial distribution shifts, and a real-world Crazyflie demonstration after an agent failure. The equivariance property is argued from the structure of the canonicalization-decanonicalization pipeline.","tokens_in":13438,"tokens_out":8859,"duration_ms":76779,"significance":"If the claims hold, LEGO offers a practical and modular way to inject geometric and permutation symmetries into standard MARL algorithms, which is valuable for swarm control. Strengths of the paper include a clean architectural decomposition, a public code repository, comparison against several natural baselines, and real-hardware validation with graceful degradation after agent failure. However, the central formal claim of E(2)-equivariant policies is not fully established for the partially observable Tag-occlusion task because the canonicalization relies on the global center of mass, which may not be available under occlusion. The absence of a true equivariant MARL baseline also tempers the comparative claims.","major_comments":[{"comment":"In Eq. (2), the canonical y-axis is defined using the global center of mass c = (1/N) Σ_j p_j over all N agents. In the Tag-occlusion task (Section V-B), each agent observes only teammates and opponents that are not occluded, so this c cannot be computed from O_i. The paper does not state whether c is formed from the full global state X or from the visible subset. If the full state is used at execution, the actor is not decentralized under CTDE and the partial-observability benchmark is not clean; if only visible agents are used, Eq. (2) is not implemented as written and the claimed E(2)-invariance must be re-derived for a visibility-dependent CoM, which is not guaranteed to hold. This ambiguity is load-bearing for the competitive Tag-occlusion results and for the paper's 'local canonicalization' contribution.","section":"IV-B, Eq. (2); V-B"},{"comment":"The formal equivariance statement in Section IV-B, C_i(ρ_E(2)(R,t) X) = C_i(X), is stated for the global state X, but the deployed policy acts on observations O_i of a partially observable Markov game (Section III). In Tag-occlusion, the observation function is defined through occlusion by obstacles, and the paper does not specify a group action on O_i that commutes with the model. Without such a specification, the 'by construction' E(2)-equivariance of the policy does not directly extend to the actual observation-to-action mapping. Please state the assumptions under which policy equivariance holds, or restrict the claim to fully observed settings.","section":"IV-B/E and III"},{"comment":"The abstract claims that LEGO outperforms 'equivariant baselines,' but the experimental comparison in Section V-A includes MAPPO, MAPPO-local, and MAPPO-GNN; no true equivariant MARL method from the related-work list (e.g., [39] or [42]) is evaluated. The MAPPO-local ablation is useful, but it does not substantiate superiority over existing equivariant architectures. Please add a stronger equivariant baseline or soften the wording.","section":"V-A and Abstract"}],"minor_comments":[{"comment":"The role-wise pooling is described as pooling over all nodes in each subgraph, which would yield the same pooled vector for every agent of the same role; please clarify whether the agent's own node is treated separately or whether agent identity is otherwise preserved, since this affects how same-role agents differentiate.","section":"IV-D"},{"comment":"The caption says the 'global observation X' is canonicalized, while the text defines the canonicalization on X as the global state; please align the notation between global state and agent observation to avoid confusion about what is available at execution.","section":"Fig. 2 caption and IV-B"},{"comment":"The introduction and abstract use E(n) while Section IV-B defines E(2); please use the dimension consistently.","section":"Introduction and Abstract"},{"comment":"The dagger symbol is used for MAPPO, MAPPO-GNN, and LEGO-MAPPO at 4 agents to indicate models specially trained for that configuration; please check whether the markers apply to MAPPO-GNN as intended, given the text states only LEGO is trained at 4 agents.","section":"Table I"},{"comment":"The real-world experiments use the global state X to track positions; please clarify whether the deployed actors receive local observations or full state, to reconcile with the partial-observability claims made elsewhere.","section":"V-D"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope for a robotics venue, but the partial-observability/global-CoM ambiguity is central to the contribution and should be resolved in revision. The lack of a genuine equivariant baseline also weakens the comparative claims. The code release and hardware demonstration are strengths that should be preserved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nShort version: this is a genuinely useful empirical paper with one load-bearing ambiguity in the competitive task, and the authors need to resolve it before I'd accept the strong claims. I'd still send it to serious peer review, because the cooperative results and the architecture are worth referee time.\n\nWhat's actually new: the specific combination of agent-centric canonicalization with role-aware Graphormer encoders inside MAPPO, plus a careful evaluation that includes cross-validation in a competitive setup and zero-shot team-size transfer. The pieces individually exist—canonicalization for equivariant RL, graph transformers, heterogeneous role graphs—but the integration is done cleanly and the ablations (MAPPO-local vs MAPPO-GNN vs LEGO) do isolate the contributions. The cooperative MPE Spread results are convincing, and the cross-validation idea is a good way to cut through the usual ambiguity of competitive reward curves.\n\nSoft spots, in order of severity.\n\nFirst, the partial-observability conflict flagged in the stress test is real. The canonical y-axis uses the global center of mass over all N agents (Eq. 2). In Tag-occlusion, agents explicitly cannot see occluded teammates/opponents, so either the implementation is using hidden global state at execution—which breaks the decentralized CTDE claim—or it uses a visible subset, in which case Eq. 2 as written is not what runs and the 'by construction' E(2)-invariance argument no longer follows. The paper needs to state which, and an ablation comparing the two would settle it. This does not sink the cooperative fully-observed results, but it directly undercuts the competitive benchmark.\n\nSecond, the baselines are weaker than the related-work section implies. The paper cites true equivariant MARL methods but doesn't run them. MAPPO-local is a reasonable canonicalization-only baseline, but an actual E(2)-equivariant network (or the equivariant actor-critic from the cited [39] or [51]) would test whether the Graphormer encoding adds anything beyond the canonicalization. Minor: the Crazyflie demo is qualitative, but I don't hold that against the paper; hardware demos in this literature are usually like that.\n\nCitation pattern looks fine. Self-citations are relevant and not padded. No circularity in the main claim.\n\nBottom line: solid engineering contribution, central claim plausible in cooperative settings, but the partial-observability ambiguity is load-bearing for the competitive results. The paper deserves a serious referee, and I'd ask the authors for a clear statement of how c is computed under occlusion plus at least one strong equivariant baseline.","headline":"A useful, well-ablationed MARL architecture whose competitive-results claims hang on an unresolved global-vs-local center-of-mass ambiguity under partial observability.","tokens_in":13833,"tokens_out":2373,"would_cite":true,"duration_ms":21623,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Canonicalizing each agent's observation before graph encoding yields an E(2)-equivariant policy that transfers across team sizes.","keywords":["multi-agent reinforcement learning","equivariant policy","canonicalization","graph neural networks","swarm control","zero-shot generalization","E(2) equivariance","role-aware graph encoding"],"falsifier":"Train the same LEGO-MAPPO architecture on the occluded Tag task twice, once with the global center of mass available for canonicalization and once with the frame built only from visible agents; if the second variant's reward or the E(2)-invariance of its actions degrades substantially, the claim of decentralized E(2)-equivariant policies under partial observability is not supported. A second check is to rotate and translate the initial configuration by a known E(2) transformation while preserving relative geometry, then verify that the policy's rendered global actions rotate and translate identically at every timestep.","tokens_in":12914,"feed_emoji":"🤖","tokens_out":7384,"duration_ms":59184,"temperature":0.7,"pith_summary":"The paper introduces LEGO, a policy architecture for multi-agent reinforcement learning, and claims that combining per-agent canonicalization with role-aware graph encoding makes learned swarm-control policies equivariant to the Euclidean symmetries of the plane and permutation-symmetric within roles. The practical claim is that with these symmetries built in, policies train much more sample-efficiently and, unlike standard MLP policies, keep working when the team grows or shrinks, when the scene is rotated or shifted, and when an agent fails. The paper supports this with MAPPO-instantiated experiments in cooperative coverage and competitive pursuit-evasion with occlusion, plus quadrotor trials.","feed_headline":"Canonicalizing each agent's view lets one policy handle any swarm size","feed_subtitle":"Agent-centric frames plus role-aware attention transfer without fine-tuning to new team sizes and survive agent failures.","key_machinery":"The load-bearing object is the agent-centric canonical frame $R_i$ built from the agent's velocity $v_i$ and the team center of mass $c = \\frac{1}{N}\\sum_j p_j$, which maps the global state $X$ into an E(2)-invariant local observation $C_i(X) = \\rho_{SE(2)}(g_i^{-1})X$. This frame does two jobs: it removes global rotation and translation as nuisance variation before learning, and, because the predicted local action is multiplied back by $R_i$, it guarantees the end-to-end policy is E(2)-equivariant. The second component is the role-aware Graphormer, a transformer-style attention encoder applied separately to dense subgraphs of each role and then pooled per role, which supplies permutation equivariance within roles and a fixed-size representation across team sizes.","core_discovery":"The central claim is that LEGO produces an E(2)-equivariant policy by construction: each agent first rewrites the global scene in its own local frame, whose x-axis is the agent's velocity and whose y-axis is oriented toward the team's center of mass; a role-aware graph transformer then encodes the canonicalized scene; the actor outputs a local action; and rotating that local action back by the same frame yields a global action that transforms correctly under rotation and translation of the whole scene. Because the encoder's output is pooled per role rather than per agent, the representation's dimension depends on the number of roles, not the number of agents, so the same policy can act for swarms of different sizes. The paper's experiments claim this yields faster convergence and higher rewards than MLP-based, graph-only, and canonicalization-only baselines, zero-shot transfer to 2 through 6 agents, out-of-distribution robustness, and continued operation after one pursuer breaks down in real-world trials.","pith_inferences":["A testable extension the paper does not report: building the canonical frame from only the visible subset of agents in the occluded task would determine whether the policy remains decentralized and E(2)-invariant when the global center of mass is unavailable.","Because each role is encoded in a separate subgraph and pooled independently, the architecture could plausibly admit an entirely new role at deployment by adding one more role-stream; the paper does not demonstrate this.","The curriculum result suggests that the warm-start benefit comes partly from reusing the canonicalization geometry itself, so the gain may grow with team size; this is an inference, not a claim the paper makes."],"forward_implications":["The same policy trained on one team size can be deployed, without fine-tuning, on smaller and larger swarms; the paper reports this for a 4-agent policy evaluated on 2, 3, 5, and 6 agents.","Euclidean equivariance removes the need to learn rotations and translations from data; the reported training curves converge faster and reach higher rewards than MLP, canonicalization-only, and GNN-only baselines.","Role-wise pooling keeps the representation size fixed as the team grows, so the architecture can be used as a drop-in actor for MAPPO and is claimed to integrate with standard MARL algorithms.","Because the policy tolerates agent removal, including one pursuer disabled mid-episode in real-world trials, it offers graceful degradation for swarm missions.","Curriculum initialization from a smaller team gives additional gains on larger teams, with the paper reporting best results when a 4-agent policy warm-starts training on 6, 7, and 8 agents."],"supporting_citations":[{"why":"Supplies MAPPO, the MARL optimization algorithm that LEGO is instantiated with and that all baselines share.","marker":"[14]"},{"why":"Supplies Graphormer, the attention-based graph encoder used for the role-wise subgraphs.","marker":"[49]"},{"why":"Introduces the canonicalize-then-decanonicalize approach to SO(2)-equivariant reinforcement learning that LEGO's local canonicalization builds on.","marker":"[9]"},{"why":"Defines the MPE Spread cooperative task and the multi-agent actor-critic setting used for evaluation.","marker":"[23]"},{"why":"Provides the Tag environment on which the occluded competitive benchmark is based.","marker":"[56]"},{"why":"Provides heterogeneous-agent role modeling that motivates the role-wise graph decomposition.","marker":"[13]"}],"fun_headline_variants":["Canonicalize per agent, act globally: swarm control that scales","E(2)-equivariant policy that transfers to any team size","Role-aware encoding plus frame canonicalization beats baselines","LEGO: one policy for any swarm size via local canonicalization","Local frames and role-wise graphs make swarm policies size-agnostic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The canonical y-axis is defined from the global center of mass of all agents, and the claimed invariance and decentralization of the policy hold only if each agent can compute that center of mass from its own observation; under occlusion the paper does not state how.","fun_headline_variants_meta":{"raw":{"variants":["Canonicalize per agent, act globally: swarm control that scales","E(2)-equivariant policy that transfers to any team size","Role-aware encoding plus frame canonicalization beats baselines","LEGO: one policy for any swarm size via local canonicalization","Local frames and role-wise graphs make swarm policies size-agnostic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1299,"prompt_tokens":966,"completion_tokens":333,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":582,"completion_tokens_details":{"reasoning_tokens":244}},"tokens_in":582,"tokens_out":333,"duration_ms":3447,"temperature":1.0,"reasoning_tokens":244,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:51:03.124322+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the same LEGO-MAPPO architecture on the occluded Tag task twice, once with the global center of mass available for canonicalization and once with the frame built only from visible agents; if the second variant's reward or the E(2)-invariance of its actions degrades substantially, the claim of decentralized E(2)-equivariant policies under partial observability is not supported. A second check is to rotate and translate the initial configuration by a known E(2) transformation while preserving relative geometry, then verify that the policy's rendered global actions rotate and translate identically at every timestep.","supporting_citations":[{"cited_title":"Pettingzoo: Gym for multi-agent reinforcement learning,","cited_arxiv_id":null,"evidence_quote":"Provides the Tag environment on which the occluded competitive benchmark is based."}],"review_version":2}