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Constrained Multi-Agent Reinforcement Learning in Imagination

Raz Shmueli, Shahaf S. Shperberg

arXiv:2610.21319v1cs.AI

TL;DR

Existing constrained multi-agent methods learn the return–safety trade-off without a model, paying for updates with costly real interactions. CA-MAMBA combines cost prediction with a multi-agent world model; at equal budgets, it outperforms model-free baselines across the reported benchmarks.

  • Problem

    Existing constrained multi-agent methods learn the return–safety trade-off without a model, so every policy update requires slow, costly interactions that can incur the cost being constrained.

    Source: §1 · p. 1

  • Method

    CA-MAMBA combines MAMBA with SafeDreamer’s cost prediction to form one team constraint from per-agent predictions, with cost priced from measurements.

    Source: §1 · p. 2

  • Results

    On 23 of 24 discrete-control configurations CA-MAMBA had lower cost than two model-free constrained baselines; on MAMuJoCo it earned 2.8 to 12 times their reward at equal interaction budgets.

    Source: §1 · p. 1

  • Takeaways & Limitations

    In controlled tests on one task per benchmark, pricing cost advantages at parity more than halved a cost-blind learner’s cost and met a tight continuous-control limit the learned multiplier missed.

    Source: §7 · p. 8

    The results do not guarantee that any episode stays within the limit or that learning is safer.

    Source: §6 · p. 8

Abstract

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Predicting cost in a multi-agent world model moves most constrained policy updates into imagination: the stronger model-free baseline needs $10$ to $20$ times CA-MAMBA's interactions on the median SMAC configuration, and CA-MAMBA earns $2.8$ to $12$ times the baselines' MAMuJoCo reward at equal budget. In controlled tests on one task per benchmark, pricing the model's cost advantages at parity more than halves a cost-blind learner's cost and meets a tight continuous-control limit the learned multiplier misses, suggesting the model's cost estimates are not the bottleneck.

1 Introduction

CA-MAMBA puts cost prediction in a multi-agent world model so constrained policy updates can use imagined rollouts. It improves interaction efficiency over model-free baselines, while controlled tests separate model cost estimates from multiplier behavior.

1 Introduction

  • CA-MAMBA predicts each agent’s cost, trains per-agent policies on imagined reward and cost, and prices the team constraint using measured cost.Its components combine MAMBA with SafeDreamer-style cost prediction.Source: §1 · p. 1
  • The paper decomposes the imagined-to-real cost gap into cost-prediction and next-step prediction biases, with the transition bound split across agents by communication structure.The decomposition motivates pricing cost from measurements rather than sizing a safety margin from model-error bounds that training data cannot usefully estimate.Source: §1 · p. 1
  • On the median SMAC configuration, the stronger model-free baseline needs 10 to 20 times CA-MAMBA’s interactions to match its cost and win rate.At equal budgets, CA-MAMBA has lower cost on 23 of 24 discrete-control configurations and a higher win rate on 22.Source: §1 · p. 1
  • At equal interaction budgets, CA-MAMBA earns 2.8 to 12 times the model-free baselines’ MAMuJoCo reward, with all three methods ending within the limit.No method meets the zero-casualty limit; CA-MAMBA meets 5 of 12 nonzero limits, which the stronger baseline matches only with ten times the interactions.Source: §1 · p. 1
  • At parity, model cost advantages more than halve a cost-blind learner’s cost, while the learned multiplier misses a tight continuous-control limit that parity meets at equal reward.These controlled experiments cover one task per benchmark; at five times parity, agents avoid combat and win no battle.Source: §1 · p. 1
  • The contributions also include controlled multiplier experiments on both benchmarks and evidence that acting without messages can cost reward under continuous control.The experiments aim to separate the price’s role from the model’s role.Source: §1 · p. 2

2 Related Work

Prior constrained multi-agent methods update policies without a model, while world-model methods move policy training to imagined trajectories. CA-MAMBA combines multi-agent world modeling with cost prediction, addressing cost and team-interaction questions not covered by those lines of work.

Constrained multi-agent learning

  • Constrained multi-agent methods optimize return under expected cumulative-cost limits, but learn without a model and pay environment interactions for every policy update.MACPO preserves a trust-region update, while MAPPO-Lagrangian subtracts a learned cost multiple from reward; shielding instead replaces actions predicted to violate safety.Source: §2 · p. 2

Learning in imagination

  • World models train policies on imagined trajectories; Dreamer lacks cost, while LAMBDA and SafeDreamer add constraints for single-agent settings.SafeDreamer’s multiplier follows imagined cost when its policy trains in imagination.Source: §2 · p. 2
  • MAMBA models teams with per-agent dynamics coupled through attention, but existing multi-agent models do not predict cost.Other designs use interacting local models or centrally aggregated per-agent transformer representations.Source: §2 · p. 2

Benchmarks

  • CA-MAMBA combines MAMBA and SafeDreamer’s cost prediction, raising questions about forming a team constraint, communication and correlation in the cost gap, and choosing the multiplier signal.The paper evaluates these issues on MAMuJoCo velocity tasks and SMAC with a constructed casualty cost.Source: §2 · p. 2

3 CA-MAMBA

CA-MAMBA models team dynamics and cost, then trains policies on imagined reward and cost while adjusting a shared cost price from measured episodes. Its design connects per-agent histories through attention messages and uses imagined actor–critic updates under a team constraint.

3.1 Constrained Dec-POMDP with Messages

  • The team’s decentralized policies maximize shared return subject to a limit on expected undiscounted episode cost, with each agent acting from its own history and optional messages.The setup is a cost-extended decentralized partially observable Markov decision process; the environment may charge agents a shared team cost or distinct individual costs.Source: §3.1 · p. 2

3.3 The Message Function and Its Neighborhood

  • Each agent’s message is produced by attention over its chosen neighborhood and carries teammates’ previous actions and stochastic states into its recurrent transition.Agents cannot condition on teammates’ current actions, and the actor has no attention of its own, making the neighborhood the only communication channel.Source: Figure 1 · p. 3, §3.2 · p. 3, §3.3 · p. 4

3.2 Cost-Aware Multi-Agent World Model

  • CA-MAMBA adds per-agent cost prediction to a recurrent state-space world model, allowing policy updates to use generated trajectories instead of collected interactions.The model learns latent dynamics from replay and uses its prior to roll forward without observations during imagination.Source: §3.2 · p. 3

3.4 Imagined Actor–Critic and Measured-Cost Multiplier

  • The imagined actor–critic uses separate reward and team-cost critics, while one nonnegative Lagrange multiplier prices cost for the team.Factorized policies can make the relaxation inexact, so the method uses it as a training signal without claiming it is tight.Source: §3.4 · p. 4
  • The multiplier rises when measured episode cost exceeds the limit and falls when it is below, while the actor’s update uses imagined, standardized reward and cost advantages.The episode cost is measured on real interactions; the imagined actor objective and measured-cost multiplier are not gradients of one objective.Source: §3.4 · p. 4

4 What Model Error Permits

The imagined-to-real cost gap is determined by cost-head and transition biases evaluated along states visited by the real system. The analysis relates transition error to agent neighborhoods and inter-agent correlations, while showing that practical bounds are often uninformative and measured-cost pricing does not fix every model failure.

4 What Model Error Permits

  • The imagined-to-real discounted cost gap decomposes exactly into current-cost bias and next-state prediction bias, averaged over states visited by the real system.A bound on those average errors yields a safety margin, but at γ = 0.99 and horizon 14 its stated form weights transition error far more than cost-head error.Source: §4 · p. 5

Where the agents enter

  • The team transition-error bound sums per-agent errors and a correlation term η, which remains even with exact per-agent predictors when agents’ next states are correlated.Each per-agent error depends on whom that agent attends to, so communication enters the safety margin.Source: §4 · p. 5

How large the gap is in practice

  • The empirical safety-margin bound is uninformative because prior–posterior divergence yields transition-error bounds above 1, and any ε_P above 0.16 makes the margin exceed c_max.The estimated cost-head bias contributes about 0.05 to the margin on MAMuJoCo at d = 25.Source: §4 · p. 6
  • On 18 runs, the model under-predicts realized cost in 17; the shortfall is 33% on 8m at d = 0 and 35% on HalfCheetah 2×3 at d = 25.The replay cost-head bias is under 1% of realized cost on 8m, while the prior’s bias has the matching sign there.Source: §4 · p. 6

Why the multiplier reads measured cost

  • Measured-cost pricing targets the real constraint despite model-bias sign, but convergence is not guaranteed and a cost head blind to unsafe actions can leave the policy unsafe.The analysis gives a long-run average-cost condition if the multiplier grows slower than linearly, but does not bound violating episodes or prove iterates converge.Source: §4 · p. 6

5 Experiments

The experiments compare CA-MAMBA with model-free constrained baselines on SMAC and MAMuJoCo, then investigate missed cost limits and the effect of acting without messages.

5.1 Interaction Efficiency on SMAC

  • On SMAC, CA-MAMBA has lower cost than both baselines on 23 of 24 configurations and higher win rate on 22; MAPPO-Lagrangian needs a median 20 times the interactions to match both at 100k.It does not match CA-MAMBA within 50 times the interactions on five configurations.Source: Figure 2 · p. 7, §5.1 · p. 7

5.2 Interaction Efficiency on MAMuJoCo

  • At the matched 1M budget, CA-MAMBA earns 2.8 to 12 times either baseline’s reward in all three MAMuJoCo environments.At that checkpoint, every method’s mean cost is under d=25 except no-communication HalfCheetah.Source: Table 1 · p. 7, §5.2 · p. 7

5.3 Why Cost Limits Are Sometimes Missed

  • On SMAC 8m, pricing cost advantages at parity cuts cost from 5.28 for the cost-blind learner to 1.75 and raises win rate from 0.63 to 0.97, while no tested rule meets the limit.On HalfCheetah 2×3 at d=5, the learned multiplier misses by a factor of 3.6, while fixed β=1 ends all six runs within the limit at similar reward.Source: Table 2 · p. 8, §5.3 · p. 8
  • At 100k interactions, CA-MAMBA’s mean cost meets 5 limits, matching MAPPO-Lagrangian at 1M and MACPO at 5M; MAPPO-Lagrangian meets 8 limits at 5M.At tight MAMuJoCo limits d=0.2, 1, and 5 on Ant 2×4, Ant 4×2, and HalfCheetah, CA-MAMBA at αβ=10^-5 exceeds all three; with ten times the interactions, MAPPO-Lagrangian meets all three.Source: §5.3 · p. 8

5.4 Acting Without Messages

  • Acting without messages leaves reward unchanged on Ant 4×2, lowers it 19% on Ant 2×4 within run-to-run spread, and reduces it from 2527 to 982 on HalfCheetah, with no consistent cost effect.Training and imagination still use messages in these runs.Source: §5.4 · p. 8

6 Limitations

The reported gains are bounded by the evaluation settings and do not establish episode-level safety or universal constraint satisfaction.

6 Limitations

  • Matched-budget results do not guarantee that no episode exceeds the limit or that learning is safer; on Ant 2×4, cumulative cost above the limit exceeds MACPO’s, and intolerant deployments would also need a shield.The reported evidence supports lower cost on SMAC or more reward within the limit on MAMuJoCo, not a guarantee of zero violations.Source: §6 · p. 8
  • The evidence covers average-case model error, shared team costs, action-time message removal, and multiplier tests on one task per benchmark with three runs per setting.The frequency of cost-head errors on actual unit-loss steps is unmeasured; individual-agent constraints and training a self-only model remain untested.Source: §6 · p. 8

7 Conclusion

The conclusion reports interaction and reward gains alongside controlled evidence about cost prediction; the appendices specify the training updates and communication pathway.

7 Conclusion

  • The stronger model-free baseline needs 10 to 20 times CA-MAMBA’s interactions on the median SMAC configuration, while CA-MAMBA earns 2.8 to 12 times the baselines’ MAMuJoCo reward at equal budget.Controlled tests on one task per benchmark show that parity-priced cost advantages more than halve a cost-blind learner’s cost and meet a tight continuous-control limit missed by the learned multiplier.Source: §7 · p. 8

A Training Procedure

  • The shared multiplier is updated from measured episode cost after collection, so that cost statistic reflects a lagged policy rather than the policy that follows.Across one learner step, the multiplier updates compose into a projected dual update.Source: A Training Procedure · p. 9

B Execution Under the Full Neighborhood

  • Each agent runs its own world model and parameter-tied actor, with neighborhood attention as the only inter-agent channel; the communication ablation removes messages only during action selection.World-model learning and imagination retain the full neighborhood, and neither setting uses privileged global state.Source: B Execution Under the Full Neighborhood · p. 9

C Verified Implementation Settings

The appendix verifies the implemented training, architecture, and compute settings, while documenting that run labels do not make individual runs reproducible.

C Verified Implementation Settings

  • SMAC runs use dead_allies_incremental, while the collision study uses dense collision cost; the discrete-SMAC MACPO adaptation is not part of the CA-MAMBA code.MAMuJoCo runs use the velocity cost of Appendix D.Source: C Verified Implementation Settings · pp. 9, 11
  • 1 is the world-model loss weight for every term, and KL balancing assigns 0.8 to prior updates and 0.2 to posterior updates.The listed terms include the divergence and MAMBA’s continuation, action-availability, and discrete-action classifier heads.Source: Table 3 · p. 10
  • 60 world-model gradient steps precede 4 policy-update rounds per collected episode; each round generates a fresh imagined rollout and uses 5 PPO epochs.World-model minibatches contain 40 sequences of length 20, and PPO minibatches contain 2,000 imagined transitions.Source: C Verified Implementation Settings · p. 10
  • Each agent runs its own world model and a parameter-tied actor, with transition attention as the only inter-agent channel.Full attention exchanges previous actions and stochastic latents; self-only attention removes that exchange, while world-model learning and imagination keep full attention.Source: Table 4 · p. 10, Figure 3 · p. 10, C Verified Implementation Settings · p. 10
  • MAMuJoCo uses a Normal policy with tanh squashing and evaluates log-probability at the pre-squash sample; otherwise the policy collapses through entropy death within roughly 55k interactions.Its world-model, actor, and critic rates are 10−4, 3 × 10−5, and 10−4, respectively.Source: C Verified Implementation Settings · p. 10
  • 4.5 hours is the reported time for 1M HalfCheetah 2×3 interactions; on 8m, 100k interactions take 16.3 ± 0.4 hours.The reported model-free baselines are faster per interaction, and timings exclude queueing.Source: C Verified Implementation Settings · p. 10
  • Run labels 1 to 3 do not seed the random-number generators, so individual runs are not bit-reproducible from their labels.The three runs are independent samples; library versions are also unpinned.Source: C Verified Implementation Settings · pp. 10–11

D MAMuJoCo Details

Across MAMuJoCo and SMAC experiments, CA-MAMBA’s interaction-budget and cost results vary by task and multiplier setting. Cost-aware components reduce casualties without lowering return in the SMAC ablation, while replay diagnostics find small cost-head bias but limitations in what the logs establish.

D MAMuJoCo Details

  • At 1M interactions, both model-free MAMuJoCo baselines remain in a low-velocity regime, so their near-zero cost reflects an undeveloped gait rather than constraint satisfaction.The tasks are Ant 2×4, Ant 4×2, and HalfCheetah 2×3; the reported baseline runs use 10M interactions and three seeds (with one noted exception).Source: D MAMuJoCo Details · p. 11
  • At tight limits, no CA-MAMBA multiplier rule meets either Ant limit at 1M, while MAPPO-Lagrangian meets all three limits at 10M; CA-MAMBA meets HalfCheetah’s limit with αβ=10−4.On HalfCheetah, that run has mean cost 4.6 and reward 1481, versus MACPO’s reward 1317 and MAPPO-Lagrangian’s 1138 after 10M interactions.Source: D MAMuJoCo Details · p. 11
  • Raising the multiplier learning rate reduces cost by 83% on Ant 2×4 and 75% on HalfCheetah under tight limits, but reduces reward by 29% and 25%, respectively.On Ant 4×2, cost does not fall; under d=25, a tenfold rate increase cuts HalfCheetah reward by 55% while reducing already-satisfied cost from 16.6 to 0.0.Source: D MAMuJoCo Details · pp. 11–12

E SMAC Learning Curves

  • On the easier SMAC maps, MACPO-Discrete catches CA-MAMBA’s win rate only after 5M interactions versus 100k; on the harder maps it stays at zero win rate.CA-MAMBA converges within 100k on the harder maps; these curves use an imagined-cost multiplier and are presented as training dynamics, not main-result measurements.Source: E SMAC Learning Curves · p. 13

F Dense Collision-Cost Results

  • With dense collision cost, CA-MAMBA at 100k has higher score and win rate than MACPO at 1M on 9 of 10 maps, but higher collision cost on 5.At 1M, MACPO wins no episode on 7 of the 10 maps.Source: F Dense Collision-Cost Results · p. 13

G Safety-Mechanism Ablation

  • Removing the cost critic and multiplier raises endpoint casualties by about two to three on 1c3s5z and about one on 8m, without a return drop.These are separate runs from the main results and are interpreted for curve shape rather than endpoint statistics.Source: G Safety-Mechanism Ablation · p. 13

H Cost-Model Accuracy

  • The replay cost-head bias’s training-average absolute value is 0.004 on HalfCheetah 2×3, 0.003 on Ant 2×4, and 0.001 on Ant 4×2.The values are upper estimates of |B_c| inflated by batch noise; the head tracks the HalfCheetah cost arc without visible lag or systematic offset.Source: H Cost-Model Accuracy · p. 13
  • The cost-head diagnostics cover continuous velocity cost, not sparse irreversible SMAC casualties, and replay batches do not certify the theorem’s state-distribution condition.The transition term cannot be bounded from these logs; the measured gap between imagined and realized cost is used as transfer evidence instead.Source: H Cost-Model Accuracy · p. 13

I Model-Error Estimates

The model under-predicts imagined-window cost in 17 of 18 measured runs, but the logged diagnostics do not certify a useful transition-error bound. The cost comparison is an average-transfer check rather than a worst-case guarantee.

I Model-Error Estimates

  • The logged prior-posterior divergence proxy yields no useful transition-error bound, even for an exact model, because it can remain positive when observations inform the latent.Its Pinsker/Jensen bound is above the trivial total-variation value of 1 in the reported logs.Source: I Model-Error Estimates · p. 13
  • The window-cost comparison spreads realized cost evenly across an episode, so it checks average transfer and understates bursty windows rather than testing worst-case transfer..Source: I Model-Error Estimates · p. 14
  • The model under-predicts window cost in every Table 11 row and 17 of 18 runs; the exception has realized window cost 0.004.Three runs per row are too few to distinguish the price rules.Source: I Model-Error Estimates · p. 14
  • On HalfCheetah 2×3 at d=25 under the learned multiplier, realized cost exceeds imagined cost by 0.020 ± 0.020 over the 14-step window, or 35% of mean realized cost.The replay-state estimates pool states and use a critic rather than the model’s own cost-to-go, so they bound neither theorem term.Source: I Model-Error Estimates · p. 15
  • On 8m at d=4, the next-state cost-to-go bias is positive under every multiplier rule, ranging from 2.5% to 5% of the critic’s cost value.For the measured-cost multiplier it is 0.069 ± 0.005 per step against a value of 1.85; the cost-head bias is unavailable there.Source: I Model-Error Estimates · pp. 15–16

J Multiplier Dynamics

The measured-cost multiplier responds to whether costs exceed the limit, while the SMAC zero-casualty runs show that rising prices do not ensure feasibility. In the continuous-control runs, the multiplier relaxes after costs return below the limit.

Multiplier trajectories

  • On HalfCheetah 2×3, β climbed to 0.64 during an early cost overshoot, then decayed after cost returned below the limit near 0.4M interactions.The trajectories were logged over the first 1M interactions at d = 25 and αβ = 10^-5, with means and one standard deviation over three seeds.Source: J Multiplier Dynamics · p. 16
  • On Ant 2×4 and Ant 4×2, β stayed at zero while the constraint was inactive, rising only when costs appeared in Ant 2×4.On Ant 4×2, cost never reached the limit and β never left zero.Source: J Multiplier Dynamics · p. 16

Cost prediction

  • On HalfCheetah 2×3, predicted team cost followed the full rise-and-fall arc of realized cost on training batches.Source: Figure 6 · p. 17

SMAC at d = 0

  • On SMAC at a zero-casualty budget, β rose while casualty cost fell, but no rule met the limit within 100k interactions.On 8m, the measured-cost multiplier ended with most of the reduction achieved by parity pricing (β = 1).Source: J Multiplier Dynamics · pp. 16–17

K Evidence Needed for the Remaining Claims

The appendix identifies open evidence gaps around learning-time cost, sparse-cost prediction, and end-to-end communication ablations. It also points to the proofs and counterexample that address the paper’s remaining claims.

K Evidence Needed for the Remaining Claims

  • On Ant 2×4 at d = 25, CA-MAMBA accumulated roughly 23k cost over 1M interactions, versus MACPO’s 93k over 10M; at matched 1M interactions, MACPO accumulated less total cost and almost no excess above d.CA-MAMBA’s cumulative excess reached roughly 14.5k, concentrated between 0.6M and 0.9M interactions while its multiplier was still rising.Source: K Evidence Needed for the Remaining… · p. 17
  • Whether learning-time overshoot is specific to reversible velocity costs remains open; reporting SMAC’s cumulative excess and violation count would test this for irreversible casualty costs.Source: K Evidence Needed for the Remaining… · p. 17
  • The continuous velocity-cost error estimate does not cover sparse, irreversible SMAC casualty costs; positive-event recall is the proposed missing measurement.The appendix also identifies area under the precision–recall curve and absolute error of 14-step cumulative predicted cost on held-out replay.Source: K Evidence Needed for the Remaining… · pp. 17–18
  • An end-to-end self-only model ablation is needed to test whether a domain requires communication and how neighborhood changes affect per-agent transition errors.The existing communication ablation masks messages only at action selection, not during world-model learning and imagination.Source: K Evidence Needed for the Remaining… · p. 18

L Proofs and Sparse-Cost Boundary Case

  • The following appendix proves the cost-gap theorem, its bootstrapped-return extension, the factorized transition error, and measured-cost dual statements, then gives a sparse-cost counterexample.Source: L Proofs and Sparse-Cost Boundary Case · p. 19

L.1 Proof of the Imagination-to-Reality Cost Gap

The appendix proves a decomposition of the gap between imagined and real cost, then extends the analysis to bootstrapped returns, factorized transitions, and measured-cost multiplier updates. A sparse-cost counterexample shows how an unobserved cost event can leave the policy unsafe despite real-cost feedback.

L.1 Proof of the Imagination-to-Reality Cost Gap

  • Theorem L.1 decomposes the imagined-to-real cost gap into predicted-cost bias and next-step remaining-cost prediction bias, averaged over states visited by the real system.Under its stated bounds, an imagination margin can certify a real-cost threshold; the result needs no Markov or latent-sufficiency assumption.Source: §L.1 · p. 19

L.2 Bootstrapped Returns

  • The bootstrapped-return corollary extends the decomposition to λ-returns from replay start states, but does not cover advantages initialized at later imagined states.Treating termination as an absorbing state folds predicted continuation probabilities into the transition kernels.Source: §L.2 · pp. 20–21

L.3 Factorized Transition Error

  • For factorized transition models, the theorem bounds joint transition error by the sum of per-agent errors plus η, with each agent’s input neighborhood determining its model.The statement holds the state space fixed; changing neighborhoods also changes the deterministic states summarized by the model, an effect left open.Source: §L.3 · p. 21

L.4 Proof of the Measured-Cost Dual Statements

  • The measured-cost multiplier update is projected stochastic gradient descent and, under stated conditions, yields an asymptotically nonpositive average cost residual.Finite fixed points satisfy dual feasibility and complementary slackness, not a convergence or policy-optimality guarantee.Source: §L.4 · pp. 21–22

L.5 Sparse-Cost Counterexample

  • When the model misses a sparse unsafe event, it assigns equal predicted cost to safe and unsafe actions, so every finite multiplier still leaves the actor choosing the unsafe action.The multiplier then diverges without correcting the policy; positive-cost recall on visited states is what keeps the cost-bias bound small.Source: §L.5 · pp. 22–23

M Full Casualty-Cost Results

The appendix documents the SMAC configuration tables and the interaction-multiple comparison used to support the benchmark results. It also lists the MACPO hyperparameter sources and overrides.

M Full Casualty-Cost Results

  • Tables 13–17 document paired SMAC comparisons and per-configuration results, including CA-MAMBA at 100k interactions against MAPPO-Lag and MACPO at listed budgets.The reported SMAC values are training-time statistics, with differing logging windows across methods.Source: M Full Casualty-Cost Results · p. 23, Table 15 · p. 24, Table 16 · p. 25
  • The median interaction multiple for MAPPO-Lag to match CA-MAMBA’s mean cost and win rate exceeds 10× in 99% of resamples and equals 20× in 97%.The median is over 23 configurations where CA-MAMBA wins; unmatched configurations count as +∞.Source: Table 14 · p. 23

N Baseline Hyperparameters

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