The Dormant Neuron Phenomenon in Multi-Agent Reinforcement Learning Value Factorization
Authors: Haoyuan Qin
Chennan Ma
Mian Deng
Zhengzhu Liu
Songzhu Mei
Xinwang Liu
Cheng Wang
Siqi Shen
Venue: NeurIPS 2024 poster (2024)
Keywords: dormant neurons; Multi-agent reinforcement learning
Primary area: reinforcement_learning
TL;DR: We studied the dormant neuron phenomenon in multi-agent reinforcement learning value decomposition and proposed a new method for recycling dormant neurons

Abstract: In this work, we study the dormant neuron phenomenon in multi-agent reinforcement learning value factorization, where the mixing network suffers from reduced network expressivity caused by an increasing number of inactive neurons. We demonstrate the presence of the dormant neuron phenomenon across multiple environments and algorithms, and show that this phenomenon negatively affects the learning process. We show that dormant neurons correlates with the existence of over-active neurons, which have large activation scores. To address the dormant neuron issue, we propose ReBorn, a simple but effective method that transfers the weights from over-active neurons to dormant neurons. We theoretically show that this method can ensure the learned action preferences are not forgotten after the weight-transferring procedure, which increases learning effectiveness. Our extensive experiments reveal that ReBorn achieves promising results across various environments and improves the performance of multiple popular value factorization approaches. The source code of ReBorn is available in \url{https://github.com/xmu-rl-3dv/ReBorn}.

The Dormant Neuron Phenomenon in Multi-Agent Reinforcement Learning Value Factorization
Haoyuan Qin Chennan Ma Mian Deng Zhengzhu Liu Songzhu Meie, Xinwang Liu", Cheng Wang Siqi Shen Fujian Key Laboratory of Sensing and Computing for Smart Cities,
School of Informatics, Xiamen University (XMU), China
KKey Laboratory of Multimedia Trusted Perception and Efficient Computing, XMU, China
'School of Computer, National University of Defense Technology, China (haoyaangin, chennanma } @stu. xmu edu. cn, (cwang, siqi shen| x edu. cn, sz mei, xinwangliu @nudt. edu. on
Abstract
In this work, study the dormant neuron phenomenon in multi-agent reinforce- ment learning value factorization, where the mixing network suffers from reduced network expressivity caused by increasing number of inactive neurons. We demonstrate the presence of the dormant neuron phenomenon across multiple environments and algorithms, and show that this phenomenon negatively affects the learning process. We show that dormant neurons correlates with the exis- tence of over-active neurons, which have large activation scores. To address the dormant neuron issue, we propose ReBorn, a simple but effective method that transfers the weights from over-active neurons dormant neurons. We theoret- ically show that this method can ensure the learned action preferences are not forgotten after the weight-transferc procedure, which increases learning ef- fectiveness. Our extensive experiments reveal that ReBorn achieves promising results across various environments and improves the performance of multiple popular value factorization approaches. The source code of ReBorn is available in https//ggghubb..omm//mmmmtttttt
Introduction
In cooperative Multi-Agent Reinforcement Learning (MARL) [1]. a group of agents whose value function approximated using deep neural networks must cooperate achieve common goal. Deep neural network is the key driving force that scales MARL for complex decision-making tasks [2]. Recently, researchers have discovered the scaling laws that deep neural network models can increase their capacity by enlarging the size of the model and the dataset. However, single-agent reinforcement learning does not obey the scaling law and suffers from network expressivity issues [3, 4].
To alleviate the network expressivity issues in single-agent RL, researchers have proposed parameter perturbing methods. Igl et al. [5] periodically resely some layers of the value network. ReSet [6] resets the last few layers of the neural network while maintaining experience in the replay buffer. ReDo [4] periodically re-initializes the input weights of some neurons and zero out the neuron's 's output weights. Albeit these methods can improve the performance of single-agent reinforcement learning, is unclear whether they work for MARL.
"Equal contribution
I Corresponding author
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Compared to single-agent RL, MARL is more challenging, including issues such as partial- observability [7] and the non-stationary of other learning agents' policies. The Centralized Training with Decentralized Execution (CTDE) paradigm [8] is widely adopted in this context. In CTDE, it is common practice to use value factorization [2, 9], which factorizes a joint state-action value function Qior into individual agent utilities Q, Each agent acts according to Qi which is approximated using a deep neural network, named the agent network. Qi are mixed through a neural network, the mixing network, to form joint value function Qtot.
In this work, we explore the reasons behind the reduction in network expressivity issues in cooperative MARL. Specifically, we study dormant neurons [4], which remain inactive with low activation levels during learning. We demonstrate that the dormant neuron phenomenon, the number of dormant neurons increases during the training process, exists in multiple popular value-based MARL algorithms (QMIX [2], QPLEX [10], DMIX [11], and RMIX [12]) across various environments (e.g.. SMAC [13], SMACv2 [14], predator prey [15]). We find that the proportion of dormant neurons increases with the number of agents, and that dormant neurons mainly exist in the mixing network. Moreover, we identify the existence of over-active neurons, whose activation score accounts for significant portion of the activation scores for all the neurons.
Typical network parameter perturbing approaches used in single-agent RL (such as Reset [5, 6] and ReDo [4]) do not work efficiently in MARL. Parameter perturbing methods, which change the weights of neurons. may lead to forgetting of learned knowledge, especially in MARL with high cooperation demands. The cooperation knowledge should not be forgotten even after parameter perturbation. We formulate a memorization requirement that the learned cooperative action preferences remain unchanged after parameter perturbation as the Knowledge Invariant (KI) principle. We theoretically show that existing approaches [5, 6, 4] cannot guarantee adherence to the KI principle. Failing to satisfy the KI principle can lead to the violation of the Individual-Global Max (IGM) principle, which is widely adopted in MARL.
We propose, ReBorn, a simple but effective method that transfers the weights from over-active neurons to dormant neurons. It periodically detects dormant and over-active neurons, and balances the weights among them. We theoretically show that ReBorn satisfies the KI principle for various value factorization approaches (e.g., QMIX and QPLEX), distributional value factorization approaches (i.e., DMIX and DDN [11]), and risk-sensitive value factorization approach (i.e., RMIX [12]). Through extensive experiments, we demonstrate that ReBorn can improve the performance of multiple MARL value factorization methods, and performs better than multiple parameter perturbing methods by effectively remembering previously learned knowledge.
2 Background
2.1 Dec-POMDPs
We consider Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs) [16] in modeling cooperative multi-agent reinforcement learning (MARL) scenarios. A Decc-PMMPP can be defined by a tuple G = (S, [ P.r, (O {o, N, 7), where A is the set of agents, S is the states set, and U4 is the action set for agent At time step each agent i chooses an action uf. forming a joint action u., leading to a state transition g-+1 ~ P(-|s', ,u') and a joint reward r In consideration of partial observability, each agent can only make decisions based on its local observation of ~ of (-ls") E O, Each agent act according to its individual policy Ty (u,(T,) based on its local action-observation history Ti (O; x Uz)*, forming joint policy π TT; . AN >. The joint policy æ has joint action-value function: Q* (St, u,) = E 1s.. [Re St, uz]. where
R₂ Eieo 7're4s is the discounted return, is the discounting factor.
2.2 Value Function Factorization
In value factorization methods [17, 2, 9, 10, 18], per-agent utilities Q; is approximated using the agent nelwork, and they are mixed through the mixer network to form the joint state-action value function Qia- For value factorization, the Individual-Globa-- criterion that ensure the consistency between local and joint optimal action selections. It is defined as follows:
(IGM) principle [9] is critical
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Definition 1 (IGM [9]) For joint state- value function Q3. TN x xu R, where T E T a joint action--bservation history and the joint action, if there exists individual state-action functions (Q; Ti x Ui + IR. RR];=: such that the following conditions are satisfied
arg max 2j (T, u) = (arg max Q1 (T1, U1), aig max QN (TN, uN)),
(1)
u
Uh
then, we can state that [Q:] satisfy IGM for Qn under т, or Qu(T, is factorized by [Qa(Ta, =
2.3 The Dormant Neuron Phenomenon
Definition 2 (a-dormant neuron [4, 19]). Consider a fully connected layer & within a neural network, where H' denotes the total number of neurons in this layer. For input distribution D. let (x) represent the activation of newron in layer under input x neuron i is defined follows:
E D. The normalized activation score of
BRED (eE))
EE (NED)H((T))
Then a neuron in layer & can defined as u-dormant if its score of a. (i.e., 0.1)
Definition 3 (a-dormant ratio [4]). The a-dormant ratio of a neural network & can be defined as follows:
Ba -a=EMo
No. Σ H
EEG
TEG
N is the count of neurons that a-dormant in layer f, H1 is number of neurons in layer L.
The dormant neuron phenomenon refers to the steady increase the dormant ratio of the neural network throughout training.
3
Related Work
3.1
Value Factorization
Value factorization approaches [20] are widely adopted in MARL. These methods construct the joint state-action value function Qror. based on individual utility Q, VDN [17] models the joint value function the sum of individual utility function, while QMIX [2] models the monotonic increasing relationship among Qtot and Qi- Qatten [18] models the relationship through using the attention mechanism. QPLEX [10] factorizes Qtot into value function and advantage function. QTRAN [9] and ResQ [21] decompose the value function into casy-to-factorized forms. For distributional MARL, DMIX [11] factorizes value function through mean-shape decomposition. A few work [12, 22] explore risk-sensitive value factorization. RMIX [12] models the monotonic increasing relationship among Qtot and the CVaR measure of each agent's distributional utility. RiskQ [22] ensures that the collection of greedy selection of risk-sensitive individual actions is equal the greedy selection of risk-sensitive joint actions.
These methods focus on modeling the representation ability and functional relationships between the joint state-action value function and individual utilities. Our work, ReBorn, is orthogonal these approaches, can be used to improve their overall performance by reducing dormant neurons.
3.2 RL neural network expressivity
In deep reinforcement learning, neural networks tend to lose their expressive power training progresses [3]. Various studies explore the loss of expressiveness from different perspectives and propose corresponding methods mitigate this issue.
Lyle et [23] show that the instability of the target can cause the network to lose expressive ability. ReSet [6] addresses early agent experience bias by periodically resetting the last layer of the neural network. The loss of expressive ability can also be attributed to over-fitting, phenomenon analyzed in depth by Kirk et al. [24] and Zhang et al. [25] within reinforcement learning.

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Figure 1: The existence of Dormant Neuron Phenomenon in Value Function Factorization Methods.
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Figure 2: (a) The MSE Loss for fitting simple Mixing Network increases with increasing number of Dormant Neurons. It indicates that dormant neurons hurt mixing network expressivity. (b) The percentage dormant neurons in QMIX mixing network with different target network update intervals. (c) The Normalized Activation Score (NAS) percentage ranking for top-25 over-active neurons in the QMIX mixing network.
To enhance generalization, researchers propose network randomization [26], convolution architec- tures [27], and soft data augmentation [28]. Researchers [29] find that the loss of plasticity is deeply connected to changes in the curvature of the loss landscape, and plasticity injection [30] is used to enhance the learning ability of neural networks for new data. D' D'Oro et al. [31] and Yang et al. [32] propose Reset Replay to improve the sample efficiency. ReDo [4] discovers that dormant neurons occur due to the instability of the target in reinforcement learning. DRM [19] finds that the dormant neuron phenomenon related to agent exploration. When the dormancy ratio high, the agent gradually cease exploration.
ReBorn is parameter perturbing method for MARL. It can effectively reduce the number of dormant and over-active neurons. Moreover, it ensures that learned action knowledge is not forgotten after parameter perturbation.
4 The Dormant Neuron Phenomenon in MARL
Dormant neurons mainly exist in the mixing network of MARL. To verify the existence of the dormant neuron phenomenon in MARL, we analyze the number of dormant neurons during the training of QMIX [2] and QPLEX [10] across multiple tasks in SMAC [13]. The percentage of dormant neurons are illustrated Figure 1, presented separately for the agent and the mixing networks. We discover that the dormant neuron phenomenon primarily occurs in the mixing network of MARL. The percentage of dormant neurons the mixing network initially high and continues to increase, while the percentage of dormant neurons in agent networks is low. This observation is consistent across various algorithms and environments as it is depicted in Appendix D.4. The number of agents in the three tasks are 3, 10, and 27, respectively. As shown in Figure (a) to (c), with the increasing number of agents, the percentage of dormant neurons increases.
Dormant neurons hurt the expressive power of mixing networks. In MARL value factorization methods, the mixing network plays crucial role integrating individual utilities into a joint value function. As shown in [9, 20, 21], the expressive power of the mixing network significantly impacts the performance of MARL value factorization methods. The expressive power of neural networks is related to both their depth [33, 34] and width [35, 36, 37]. We study expressive power of mixing networks from the perspective of dormant neurons, We use mixing network with Multi-layer Perceptron (MLP) layers, and fit it to simple value function. This network consists of 4 neurons, and

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Figure 3: Over-active neurons in QMIX mixing networks: (a) The percentage contribution of the number of dormant neurons (depicted as Dormant), the number over-active neurons (depicted as Overactive- Number), the sum of NAS (depicted as Overactive-Sunr (eram) for over-active neurons over time. (b) Overlap coefficients for Dormant/Ovv veeraatvvvee- neurons between the current iteration and previous iterations. (c) Percentage of dormant neurons that re-enter dormancy after ReDo within different time steps.
we change the number dormant neurons from 0 to 3. As depicted in Figure 2 (a), with increasing dormant neurons, the mean square enror (MSE) loss that fits the target value increases. This indicates that an increase dormant neurons leads to reduced expressive power. Please refer to Appendix D.3 for details.
TD target non-stationarity exacerbates dormant neurons in MARL. The TD target in reinforce- ment learning is non-stationary(38)) In the MARL training process, target networks for mixing networks are typically used to stabilize TD targets. We study the impact of target non-stationarity by varying its update interval, where smaller interval indicates greater non-stationarity. As analyzed in Figure 1, the dormant neuron phenomenon primarily exists in the mixing network, so we only control the target network of the mixing network, and the comparison focuses on the dormancy ratio in the mixing network. Experimental results for the QMIX method are presented in Figure (b). As depicted, with a smaller update interval, the ratio of dormant neurons increases, indicating that increased non-stationarity in the target network results in a higher presence of dormant neurons the MARL mixing network.
The presence of over-active neurons correlate with dormant neurons. Through careful inspection of the neurons during MARL network training, we observe interesting phenomenon that has not been discovered before: some neurons exhibit very large normalized activation scores (NAS) throughout the training process. We study the percentage contribution of the average NAS of each neuron to the total average NAS of all neurons. To this end, we examine such percentage in the last layer (with 64 neurons) of QMIX's mixing network in 27m_ fsm_v____ vsm30m from SMAC. Figure 2 (c) depict the top 25 neurons which have the largest percentage. Neurons whose percentage is over 5% is depicted in red, neurons whose percentage are too low not plotted, while the other neurons plotted in green. The results show that most NAS are concentrated on a few neurons, while the NAS of other neurons are relatively low. We refer to these neurons with large NAS as over-active neurons, and define them as follows.
Definition (over-active neuron). neuron i is an over-active neuron its score s 2 B (i.e., 3).
In Figure 3 (a), the percentage contribution of the numbers of dormant neurons to all neurons, and the percentage contribution of the number over-active neurons to all neurons, along with the percentage contribution of the sum of NAS for over-active neurons to the sum of NAS of all neurons, are depicted red, blue and green, respectively. We find that, albeit there are only few over-active neurons, their NAS takes up large percentage of the neural network's 's NAS. This percentage increases steadily with the training process, correlating with the increase dormant neurons. As the percentage of the over-active neurons' NAS continues to increase, the percentage of NAS for the other neurons decreases. We conjecture that the presence of over-active neurons impacts the existence of dormant neurons.
Dormant/Over-active neurons remain dormant/over-active. To study the impact of over-active neurons on dormant neurons, we examine the percentage of dormant/over-actin neurons that remain dormant/over-activv.. As depicted in Figure (b), there a significant overlap among dormant/over- active neurons. The presence of over-active neurons appears to be significant factor contributing to the dormant neuron phenomenon, which has never been considered in previous studies. We use parameter perturbing method [4] periodically recycle the dormant neurons. Then depict the
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Figure 4: The procedure of ReBor neurons. The weights of over-active neurons are distributed randomly picked dormant neurons.
percentage of dormant neurons that re-enter dormancy within 0.2 Million steps and 0.6 Million steps in Figure 3 (c). As is depicted in the Figure, there is still significant overlap among dormant neurons. This indicates that parameter perturbing methods (such as Redo [4]) may not work efficiently for MARL, it does consider over-active neurons. We conjecture that this may be due the fact that methods developed for single-agent RL may change the neural network weights regarding agent cooperation, which lead to forgetting learned cooperative knowledge that encoded in neural network.
The ReBorn Method
In this section, we describe the Knowledge Invariant Principle, which ensures that learned action preferences do not change after perturbing neurons. We show that methods failing to satisfy this principle could lead the violation of the individual-Global-e for MARL. Then, we present the ReBorn method, which satisfies the KI principle. It balances the weights among dormant neurons and over-active neurons for the mixing network.
(IGM) principle, which is important
5.1 Knowledge Invariant Principle
Multi-agent Reinforcement Learning suffers from the dormant neuron phenomenon and the existence of over-active neurons which make the learning process inefficient. Researchers have proposed several methods [4. 5, 6] that change the weights of neurons. However, these methods overlook the complex interactions among multi-agents, and their learned knowledge may be forgotten after perturbing neurons. We formulate the memorization requirement for learned cooperation knowledge after neuron perturbations as the Knowledge Invariant Principle, which is defined follows.
Definition (Knowledge Invariant Principle (KI)). A joint state-action value function is represented as Dioe 6 (T, u) = fo(Q) (T) u1), 2% "% (TN, uN)), where fe is the mixing function that mixes Q: into Qtor. T joint observation action history, u [u,, [NN the joint action of multi-ac agent, g R + IR is a function that maps weights in to 0 g() = 0. h R+ R. h(o) -. h map the weights in & to 6. If the following condition holds:
Qioi (T. u) 2 Quo (T, = (T. 2 a. (T, u'), EIK uk + v/g
(4) then, the two functions g and h satisfy the Knowledge Invariant Principle for Quo. where [Qr(T;; w is individual agent utility function, is the number of agents, T; and u; are the observation-action history and action of agent i, respectively. 31 represents the concept of unique existence.
Given two functions and l which satisfy the KI principle, if we use them to change the joint state-action value function 0 to 2. 3lk: ug * w/z the learned knowledge that u is preferred over u' before applying g and h does not change after applying the two functions. With the KI principle, we show the following theorem.
Theorem 1. Parameter perturbing methods that do not satisfy the Knowledge Invariant (KI) principle cannot guarantee adherence to the Individual Sedividull-Globl-- Global-Max (IGM) principle.
We have theoretically shown that parameter perturbing method that does not satisfy the KI principle could lead to the violation of the 1GM principle, which the most important principle in MARL value factorization methods [2, 9, 20, 10). Furthermore, we theoretically show that two state-of-the-ar RL parameter perturbing methods, Redo [4] and ReSet [5], do not satisfy the KI principle, as detailed in Theorem and Theorem 3, respectively. These theorems and proofs are detailed in Appendix B.

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5.2 ReBorn: a Weight Sharing Method among Dormant and Over-active Neurons.
To address the issues caused by the dormant neuron phenomenon and the existence of over-active neurons, which reduce network expressivity, we propose ReBorn, a simple but effective method that shares the weights from over-active neurons with dormant neurons.
ReBorn uses an identity function h())- 0 to map the parameters of agent networks themselves, and uses function g(0) to perturb the parameters of mixing networks. The process of g(0) is described as follows. For each over-active neuron 2, we randomly select M dormant neurons that belong to the same layer as æ. Here, M is an random integer between 2 to 5. The selected dormant neurons, indexed by will share weights with neuron z. After weight sharing, these neurons will be selected again. We denote wx as the input weights for neuron z. b as the bias of neuron w as the output weights. The main procedure of the ReBorn method is depicted in Figure 4.
The input weights of dormant neurons w.." are reborn as 3, and the input weight of the over- active neuron x becomes Bo The output weights for neuron x and are reborn as no 0 and 3. a,wuut The biases for the over-active and dormant neurons
Baba, B.b.. MM
are sampled between 0.5 and 1 1.5. They are used to ensure more variation among neurons. [a:) is obtained through sampling M from normal distribution, and then Softmax operator is performed on them ensure Di ag = For dormant neurons that are not selected, we use Xavier initialization to reset their weights.
Although ReBorn is simple, have theoretically demonstrated that it satisfies the KI principle for QMIX through the following theorem.
Theorem 2. ReBorn satisfies the KI principle for the QMIX [2] value factorization method.
Moreover, we have theoretically shown that ReBorn satisfies the KI principle for value factorization method: QPLEX in Theorem 4, a distributional value factorization method DMIX in Theorem 5, and a risk-sensitive value factorization method RMIX Theorem 6. Furthermore, we show that after using ReBorn, the value functions Qto learned by QMIX, QPLEX, DMIX, and RMIX still satisfy the IGM principle in Corollary 1 to 4. These theorems and proofs are listed in Appendix B.
6 Empirical Evaluations
In this section, we present experimental results and discuss their implications. We begin with brief overview of our experimental setup in Section 6.1. Subsequently, we examine ReBorn's s robust applicability to various MARL value factorization algorithms Section 6.2. Furthermore, we demonstrate that ReBorn outperforms other parameter perturbing methods that are extended to MARL in Section 6.3. Lastly, we conduct series of ablation studies in Section 6.4. All detailed experimental results can be found in Appendix D.4.
6.1 Environmental Setup
Environments. our experiments, we employ three distinct environments that challenge the coordination and adaptability of MARL algorithms. Predator-prey simulates grid world where multiple predators collaborate to capture preys dispersed throughout the map. A successful capture requires at least 2 predators to execute the capture action simultaneously, posing a great challenge for the algorithm's 's coordination ability. The StarCraft Multi-Agent Challenge (SMAC) [13] is popular benchmark used extensively in MARL, where multiple ally units controlled by MARL algorithms aim to defeat enemy units controlled by the game's 's built-in AI. SMACv2 [14] features units that are randomly generated and positioned, enhancing stochasticity and significantly increasing the complexity of the scenarios. Please refer to Appendix D.2 for detailed descriptions.
Baselines and training. ReBorn, parameter perturbation mechanism, is applicable to various value factorization algorithms. We select classical algorithms with different types: QMIX, QPLEX, DMIX and RMIX. ReDo, ReSet, SR [31] and MARR [32] four common parameter perturbation methods in deep RI., are adapted to MARL variants to serve as baselines. Detailed implementations and parameter configurations for each algorithm are available in Appendix D.I.
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Figure 5: ReBorn can improve the performance of various value factorization algorithms: (a-h) the test win rate for the 3s57_ $esy__s__se____ f__36 and the MMM2 environments, (c) the return for predator-prey small environment, (d-f) the dormant pereent for the the 3s5z vs. 3s6z, the MMM2, and the predator-prey small environment.
6.2 ReBorn can improve the performance of various value facttoizationn algorithms
In this section, we investigate the applicability of ReBorn through validating ReBorn's ability to enhance performance across various value factorization algorithms (QMIX, QPLEX, RMIX) in different experimental scenarios (3s5z_ vs 3s6z, MMM2, predator-prey prey small). According to the experimental results presented in Figures algorithms and effectively reduce the dormant ratio of the mixing networks in diverse settings. More detailed experimental results can be found in Appendix 1 and Appendix D.4.5.
ReBor can improve the performance of multiple
6.3 ReBorn is superior to other RL parameter perturbing methods
We explore the superiority of ReBorn by applying different parameter perturbing methods to QMIX across various experimental scenarios (MMM2, 27m_vs_3000 vm_00m predator-prey large). We added ReDo, Reset, SR and MARR for comparison and further analyzed the dormant ratios and the over-active sum ratios. ReDo and ReSet are common parameter perturbation methods in deep RL, while SR and MARR are reset replay methods. All of them are adapted MARL variants to serve as baselines.
The results depicted in Figure 6 illustrate the win rates, the dormant ratios and the over-active sum ratios (the ratio of the of normalized activation scores of over-active neurons the total of scores of all neurons) across different scenarios. The results indicate that compared to ReDo and ReSet, ReBom can further enhance algorithm's performance and effectively reduce both the dormant and over-active sum ratios of the mixing network. Please refer to Appendix D.4.4 for more experimental results.
6.4 Ablation Study and Discussion
6.4.1 Satisfying the KI Principle is of great importance
In this section, we demonstrate the importance of adhering to the Knowledge Invariance (KI) principle. Our analysis in Appendix shows that applying ReBorn only to the mixing network adheres the KI principle, while using it on the entire network results in violation. We compared the performance of value factorization algorithms under the MMM2 scenario SMAC, focusing on those that either adhere to or violate the KI principle. As illustrated in Figure 7, maintaining KI with ReBorn enhances the performance across baseline algorithms, whereas violating leads to performance drop in QMIX and QPLEX, highlighting the great importance of satisfying the KI principle.
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Figure 7: Importance of satisfying the KI Principle for (a) QMIX, (b) QPLEX, and (C) RMIX. A variant of ReBorn without satisfying the Principle is depicted as Reborn w/o KI.
6.4.2 ReBorn is better than other methods that satisfy the KI principle
In this section, we explore various forms of the weight perturbation function g()) in ReBorn based on QMIX, while keeping the function h0)) = 0 constant. In ReBorn, g(0) transfers weights from over- active neurons to dormant neurons. In ReBorn (ReDo), g(0) periodically re- re-initializes the weights of dormant neurons. In ReBorn (ReSet), g periodically resets the parameters of the last layer of the neural network. In ReBom (Reverse ReDo), g()) periodically resets the input and output weights of over-active neurons. In ReBor (Pruning), (0) periodically prunes dormant neurons. To ensure the accuracy of our conclusions, we conducte experiments across various value factorization algorithms. The experimental results shown in Figure 8 demonstrate that compared with other methods, ReBorn significantly enhance the performance of value factorization algorithms.

---

MMM2
M14M2
MMM2
0.8
Ghuat suller (Raiter))
Sullear (Ralhe))
Qlicc Bollorn (Prudre))
Tea Win BaKo
0Pua0 Kudlare (Pruatre))
Test Win Rate
0.4
0.4
0.6
0.A
3:30
Environmental Staps
Covironmental Stops
Envirommentll Staps
Figure 8: Comparison with other methods that satisfy the KI principle for (a) QMIX. (b) QPLEX, and (C) RMIX.
MMM2
(b) MMM2
(c) MMM2
OWec Rullern
0HID Bollear
OMOC duffiem -224
0MS5 Butber
M
Toxt Winn Baato
Test Win Bale
Test Win
0.4
0.2
0.2
0.2
603K
8.0kk
3,200 Steps
3.6M
2M
400K
BOoK
1,20
1.CH
2M
401K
BOOK
3.20
1.0M
2M
Steps
Figure 9: Ablation different hyperparameters in ReBorn, (a) the dormant threshold a. (h) the over-active
threshold 3. (c) the ReBorn interval T.
6.4.3 Sensitivity analyses of hyper-parameters
The ablation study in Figure 9 illustrates the impact of different hyperparameter settings in QMIX- ReBorn, focusing on the ablation of the dormant threshold the over-active threshold 3, and the ReBorn interval T' The default configuration of QMIX-ReBom is a 0.1, 3 = and T 0.2.M. We modify each hyperparameter individually, and the experimental results indicate that appropriate hyperparameters help better balance network activation, thereby enhancing overall performance.
Conclusion
In this work, identify the dormant neuron phenomenon in Multi-Agent Reinforcement Learning (MARL) Value Factorization. Such a phenomenon mainly exists in the mixing network, which hurts its expressive ability. We discover the existence of over-active neurons, which correlate with dormant neurons. Existing parameter perturbing methods do not work efficiently for the dormant neurons in MARL, due to the ignorance of over-active neurons and the forgotten of learned knowledge. We formulate the memorization requirement for learning agents' cooperation knowledge as the Knowledge Invariant (KI) principle. In this work, we propose ReBorn, which is simple but effective parameter perturbing method. We show that satisfies the KI principle and can improve the performance of multiple value factorization methods better than other parameter perturbing methods.
Acknowledgement This work was partially supported by the Fundamental Research Funds for the Central Universities (No. 20720230033), by PDL (2022-PDL-12), We would like thank the anonymous reviewers for their valuable suggestions.
10

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Appendix
A Background
A.1 Dec-POMDPs
We consider Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs) [16] in modeling cooperative multi-agent reinforcement learning (MARL) scenarios. Dec-POMDP can be formally described by the tuple G = (S, {4 P.rr, [...P.PPr {O:)=- NN where N
represents the set of agents,
a finite set of states, and U4 the set of actions available to agent i. At time step t, each agent i chooses an action u E U4 forming a joint action u' E u Uh x ... x Ux. This leads to transition to a new state 5
P(-|s', u²) and joint reward rt.. In consideration of partial observability, each agent can only access an individual observation of E O,, which drawn from of (-(-(s)) y denotes the discounting factor. Each agent acts (O; x U.)" represents agent's s local action-observation
base on individual policy n., (1,44T;)) Ti), history. the global action-observation history is denoted as T E T := TI X .... X TN, on which conditions the joint policy =< T....., ... TT >. The joint policy π has joint action-value function: Q (st, uz) Exy 1:
1. 1:00 [R 8z, Uz]. where R. Ei=o y'r2+i is the discounted return.
A.2 Value Function Factorization
For cooperative multi-agent reinforcement learning tasks with partial observability challenges, agents are supposed to select actions solely based on their local observations. This presents significant challenges to global coordination in scenarios where communication is unavailable. To efficiently solve this problem, centralized training with decentralized execution (CTDE) was proposed popular paradigm. During centralized training, access to global information is available, while only local action-observaatio histories are accessible during decentralized execution phase. Value factorization is class of effective value-based methods under the CTDE paradigm, where agents make decisions based on individual utility functions. The mixing network is employed during training to fit the relationship between the joint value function and individual utility functions. Among all value factorization methods, the Individual-Glooal-- criterion that must be adhered to, ensuring the consistency between local and j optimal action selections. The definition of the IGM principle as follows:
(IGM) principle proposed by [9] a critical
Definition 6 (IGM). For a joint stale-action value function Qj: TNN Xu RR where TNN is joint action-obe obbeevvttonnvnnonn history and u is the joint action, if there exists individual state-action functions [Q: T x U4 + RR such that the following conditions are satisfied
arg max Qn( u) = (arg max QI(T1, .... max QN (TN, uN)),
(A.1)
then, we can state that [Q.I satisfy IGM for Qj under t, or Qi((T, u) is factorized by [Q,(T, W
In recent years, ensuring the adherence to the IGM principle, a series of value factorization methods have been proposed. VDN imposes additive constraints on the mixing network, and QMIX enhances VDN's representation ability by imposing monotonicity constraints. These constraints are sufficient conditions for IGM, limiting the representational ability of the joint value function. QTRAN transforms the IGM principle into linear constraint and proposes an easily factorizable form. Qatten uses the attention mechanism to model each agent's 's impact on the global situation. QPLEX decomposes the state-action value function into state value part and an advantage value part. ResQ converts the joint value function into the sum of a main function and a residual function, deriving optimal policy through masking.
13

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B Principle and Theorem
Theorem 1. After the ReBorn process, the learned value function of the QMIX [2] value factorization method still satisfy the Kl principle.
as aQ
Qioe (T. u)
C. (N(N,, Va(T)
2
(B.2)
h(w)= w, Vw E Φ h is an identity functiin
(B.3)
Bru
input weights of dormant neurons wi" input weights of over-active neurons w ouiput weights of dormant neurons w By aou
Bow
output weights of over-active neurons w
g(w)
(B.4)
Bob
bias of over-active neurons b
Beb Xavier(w)
bias of dormant neurons b; weights of non-select dormant neurons otherwise
where (T, fo(Q (T,, un), Q (N)) is the joint state-action value funtion, fo is the value factorization function of QMIX. In Reborn, 9 maps the parameters @ of the mixing network and the parameters & of the agent network to
Proof. Through the use of non-negative activation function (e.g. absolute) and hypernet [39], QMIX can ensure the following property.
S
0
v0 monotonicity property
(B.5)
aQ
indicates that if O increase, then the value of increases.
Que (T, u) u (T, u)) Elk us v/
(B.6)
Siu (T, [u1, uN]) 2 0 (T, [u],,.,, u'yl) expand a
(B.7)
fo( uz)..., ON (TN, uN)) Vo(T) fo(Q) (T,, u1() ((),,.. 22 (TN, Vo(T)
(B.8)
(o(Q9(T.. un) 22 TN,, un)) 2 fo ( ...,... 22 (TN, u'x))
(B.9)
Q (k: ux) (T), u4) 3lk uy # uf because of (B.5)
(B.10)
(T, (T, (NN,, Vg((T) 2 l2( (T) u/ Q% u'x)) + Vg((T)
u1) (N uN)) f6 (TI, u1)
u'y))
(B.11)
(B.
0.6 d (T, u) 2 Q.. (T, u')
(B.13)
(B.9) to (B.10) is because u; = u/, Vi # and 4n + ut and the monotonicy conditions. (B.10) 10) to (B.11) is due to the monotonicy condition, because that after the ReBorn process, the learned action preference of QMIX does not change.
u/, Vi + i, and Ug * 2/. Thus, we show
Corollary 1. After the ReBorn Process, the value function of QMIX remain satisfies the IGM principle.
Proof. To prove this Corollary equal to prove that the maximal action remain the same after the ReBorn method. It is shows that the ReBorn method satisfy the KI principle for QMIX, thus the
14

---

following condition is satisfy.
Pioe(T, (T, u) 2 Pio o (T, u²) - 1 (T, u) Qua0 (T, u²) Elk us # 1/2
(B.14)
Qua (T, ū 2 Vine (T, u')
á = arg max Qia! (T, u). Yuu'
(B.15)
ù = [ uN]
ii = arg max ( (Te, u.) IGM Principle
(B.16)
2i (T, ū) 2 0000 (T, u') + Quoe (T, ū) Qiae( (T, w') Vu'
KI Principle
(B.17)
Quae (T, ú + (T, u'), V'u'
(B.18)
arg max Q (T, u) arg max Quoe (T, u.)
(B.19)
(B.19) shows that the IGM principle is still preserve after applying ReBorn on the joint state-action value Qtot of QMIX.
Theorem 2. ReDo [4] with function and h does not guarantee satisfying the KI principle for the QMIX [2] value factorization method which is defined as follows.
0f
(T,
fo(Q (1, u1). (N(TN, uN)) Vo(T)
2 0
(B.20)
aQa
w E Pg E 0g otherwise
output weights of dormant neurons inpul weights of dormant newrons
g(w)
Xavier(w)
(B.21)
w
{o
E o2 E o' otherwise
outpul weights of dormant neurons imput weights of dormant neurons
h(w)
Xavier(w)
(B.22)
where fe is the value factorization function of QMIX, 9 map the parameters 0 of the mixing network to 0, h map the parameters á of the agent network to é. Xavier(w) indicates the Xavier initialization function, 82. Oa are the output/input weights of dormant neurons in 9, respectively, & are the output/input weights of dormant neurons in 6, respectively.
Proof. We prove this theorem by providing an example that ReDo does not satisfy the KI principle.
We assume that the mixing network, parameterized by f is three layer neural work. As it is depicted in Figure the input layer neurons are used for joint state-action history т. It also takes actions u as input. There are in total four joint actions represent as u = [0, 0J, u² [0. 1], u = [1,0), u² [1, 1]. There are two agents, each has two actions represent as 0 and 1, respectively. The action the first/second agent is fed into the second/third neuron of the input layer. The weights of each neurons are marked on the edges and we assume bias According to the definition 2 and the weights, the blue neuron is dormant neuron. Assuming 1. we can obtain the relationship of Q(T, u) corresponding to each action:
Q(T, u²) Q(T, u²) Q(T, u²) Q(T, u²)
(B.23)
For the dormant neuron (colored), ReDo B.22 reinitialized the input weights of the neuron using Xavier initialization, the output weights of the neurons are set to zero. The weights after ReDo are as depicted in the right part of Figure 1, and we assume that the bias for each neuron is zero. We can obtain the relationship of Q' (T, u.) corresponding each joint action:
Q 2² Q(T, u²) Q(T, u²)
Q(T, u²)
(B.24)
Since the original optimal action 3 in B.23 is different from the one u in B.24 after Redo, we can draw conclusion that ReDo does not satisfy the KI principle.
15

---

QLI (o,0
1.011)
(0,00)))))
1.000
Q(T. las, az])
Q0, I]
Q (T. (2l)
(0.1
Q(1 0]
(1,
(0. 1,1
2.00
I
Yusr
H
ReDo
Yoze H
H
2,02
001
Dormant neuron
Xovier Initialization
T
action
action
to zero
Figure 1: example to show ReDo not satisfy KI principle.
Theorem 3. ReSet [6] with function and h does not guarantee satisfying the KI principle for the QMIX [2] value factorization method which is defined follows.
000 (T, u) (TI QN (TN, uN)) Vi(T)
(B.25)
aQ
Xavier(w) 0 weights of the last layer of neural network
g(w)
(B.26)
w Xavier(w)
other layers
{
w w E weights of the last layer of neural network
h(w)
(B (B.27)
other layers
where fo is the value factorization function of QMIX, g map the parameters 0 of the mixing network to 0, h map the parameters d of the agent network to b. Xavier(w) indicates the Xavier initialization function.
Proof. We prove this theorem by providing an example that ReSet does not satisfy the KI principle. Consider a three-layer mixing network parameterized by 0, which takes joint oetion-observation history, represented as and actions as input, shown in Figure 2. There are in total four actions represent as u² [0, 0), u² [0, 1], u [1, 0]. a [1, 1]. There two agents, each has two actions represent as 0 and 1, respectively. The action of the first/second agent is fed into the second/third neuron of the input layer. The weights of each neurons are marked on the edges and we assume bias 5 = 0. Assuming T = 1. we can obtain the relationship of Q(T, u) corresponding to each action:
Q(T, u²) > Q(T, utt) > Q(T, u²) > Q(T, u?)
(B.28)
The weights of the last layer are reinitialized using Xavier initialization according to B.27. The weights after ReDo are as depicted in the right part of Figure We can obtain the relationship of Q' (T, (T,u) u) corresponding each joint action:
Q(T, xn Q(T, Q(T (1,,u²)
Q(T,u!)
(B.29)
Since the original optimal action n in B.28 is different from the optimal action w in B.29 after Reset, we can draw a conclusion that Reset does not satisfy the KI principle.
16

---

Qu (0,
(0,0.0000)
1.1
Q(T. (os, az])
Q0, I])
q' (2),))
(0,
1.9
QG. 01)
Q'0.).00)
2.2
00 D,3
(1,))
3
Ewn
Euss
ReSet
Eusz
Dus
I
Xovier Initialization
action
action
Figure 2: An example to show that Reset does satisfy Kl principle.
Theorem 4. ReBorn with functions and h satisfies the KI principle for the QPLEX [10] value factorization method.
2he (T, u) V-l,((7) (T) + 10.0 (T, u.)
(B.30)
0ʃ 20
1 (T, u) fo(A (.),..., A (TN, (T,, uN))
(B.31)
045
Q u;) (T u:) (T:) (7((7:)
max Q8 u4)
(B.32)
Vill (T) max (Io((T u))
(B.33)
input weights of dormant neurons with input weights of over-active neurons w output weights of dormant neurons w output weights of over-active neurons w bias of over-active neurons b bias of dormant neurons b; weights of non-select dormant neurons otherwise
Bows
To ag
g(w)
(B.34)
Sobe
Beb Xavier(w)
h(w) w, Ve E ④ h is an identity function
(B.35)
where fo the value factorization function of QMIX. In Reborn, 9 map the parameters 0 of the mixing network to
Proof. QPLEX uses non-negative weighted attention network implement fo which satisfies the following property.
a.a 2 0
Vi, AO, Vo monotonicity property
(B.36)
0A
17

---

Pio (T. u) 2 Q (T, u') Elk: # u
(B.37)
Vioe((T)
tot
0.0
(T, u) 2 Veg (T) + A +Li0((TT))) (T, u')
(B.38)
Viot (T) + Aloe 6 (T, [11, uN]) 2 v (r) + A (T, [4], ul
(B.39)
Vol ( [o(Af( (TI, (TI,21), A uN)) V (r) (o (TI, u/g ) A (TN
(B.40)
[0(A (TN, uN)) (T) u., 2 (TN,
(B.41)
A uz)) 2 (T u/²))
(B.42)
fT(A9(T), .... A (TN, uN)) f/(49( (TI, u/) ... A (TN, uxx))
(B.43)
v (T) + (-
(1),........ 0% (TN, uN)) > V/l (T) + (T) (-) (TN,
Q2
ux))
(B.44)
Q (T, u) 2 a..0 (T, v')
(B.45)
(B.41) to (B.42) is because u, = u/, vi kk and uy + u and the monotonicy conditions. (B.42) to (B (B.43) is due to the monotonicy condition, because in Vi + i and uy + u/g. Thus, we show that after the ReBorn process, the learned action preference of QPLEX does not change.
Corollary After the ReBorn Process, the value function of QPLEX remain satisfies the IGM principle.
Proof. The proof is the same as the proof for showing after the ReBorn process, QMIX satisfies the IGM principle. is omitted for brevity.
Theorem 5. ReBorn with functions g and h satisfies the KI principle for the DMIX [10] value factorization method. DMIX is a distribution MARL algorithm which models the distributional return Zior of aulti-agent&ee system.
2 (T, u) = Z (T, u) + + shape (T, u)
(B.46)
0f
0.00 (T, u) = 2iieon (T, u) -sa(Q ui),.., ". (NN, Vo(T)
2
0
(B.47)
aQ
Q (T;, u:) =E|Z (Ti, u:)] expectation over possible outcome of Z,
(B.48)
N
shape (Ti, (T)
(E (Ti, u.) (T, u:)) (T))
(B.49)
i=1
h(w)) w,
Vw E &
h is an identity function
(B.50)
input weights of dormant neurons wit" input weights of over-active neurons w output weights of dormant neurons w output weights of over-active neurons wout bias of over-active neurons b bias of dormant neurons b; weights of non-select dormant neurons otherwise
3ran wi
wour
Bo wout Boby
g(w)
3,0 Xavier(w)
(B.51)
In Reborn, g map the parameters 0 of the mixing network 0.
18

---

Proof.
Qu0e (T, u) 2 2oo (T, u²) Elke + us
(B.52)
Sioe (T, [u1, uN]) 2 Ph (T, (4) u/vl) expand a
(B.53)
(2) (N,, uN) + Vo(T) 2 fo(Q 1/ w/.....
+ Vo(T)
(B.54)
(T) u1). 2 TN, uN)) 2 u4). (N,, uxx))
(B.55)
2 (Tx: ux)) 2 (T., uf)) m us # u
(B.56)
(6( u1), 2% (NN, uN)) 2 f6( u/) (NN u
(B.57)
(0( (N uN + (T) 2 /0(0 (TI, 11 QN uxx) + VT((T)
(B.58)
8. 0 (T, u) 2 Qiui (T, u')
(B.59)
Thus, we show that after the ReBorn process, the learned action preference of DMIX does not change.
Corollary 3. After the ReBorn Process, the value function of DMIX remain satisfies the IGM principle.
Proof. The proof the same as the proof for showing after the ReBorn process, QMIX satisfies the IGM principle. It is omitted for brevity.
Quantile functions (inverse CDF) 0 of random variable Zs defined follows.
0z(a) inf( E R. w CDF>(z)), Va E [0,
(B.60)
where (2) is the cumulative distribution function of We denote (z(() as ((w) for simplicity. Definition (Conditional Value at Risk(CVaR)).
CV (laaa) EZ|ZZ/ S 0(a)]
(B.61)
where a is the confidence level (risk level), (()) is the quantile function (inverse CDF) defined in (B.60). CVaR is the expectation of values z that are less equal than the o-quantile value (8(a)) of the value distribution.
Theorem 6. ReBorn with functions and h satisfies the KI principle for the RMIX [12] value factorization method. RMIX is sen-isens MARL algorithm which consider risk in multi-agent system. Its joint state-action value function is defined as follows.
0f ac
2 (T, u) fo fo (TI,
(N(TN,UN)) Vo(T)
2 0
(B.62)
C (T, uz) CVaRalZ, (T,, 4:]
(B.63)
h(w) w, Ve E ó h is an identity function
(B.64)
Beuth Sowin
input weights of dormant neurons wi!" input weights of over-active neurons w output weights of dormant neurons wont output weights of over-active neurons wout bias of over-active neurons b bias of dormant neurons b; weights of non-select dormant neurons otherwise
8. aqwour
8o agw
g(w)
(B.65)
Bobz
Bebr
Xavier(w)
In Reborn, 9 map the parameters 0 of the mixing network 0.
19

---

Proof.
Pio (T. u) 2 Qi (T, bik: # uie
(B.66)
(T) (. Vo(T) 2 fe(Ci (TI, ..,, C (TN, u
Vo(T)
(B.67)
(TN,, 2 fo(C]
(N,,
(B.68)
(Tt uz)) (Ta, uk)
(B.69)
un), CN(TN, uN)) 2 lo(Cf (m. " (11 (N(TN, (T)
(B.70)
(C CNN,, uN)) (i(T) l/(C/ (TI (TN, Va(T)
(B.71)
Qiol (T, u) Q.of (T, u')
(B.72)
Thus, we show that through ReBor, the learned knowledge about action preference of RMIX does not change.
Corollary 4. After the ReBorn Process, the value function of RMIX remain satisfies the IGM principle.
Proof. In RMIX, each agent acts greedy according C2 (Ti, u:) CV [VRR, (T, uz)]. It could be viewed as Qhah (Te, u,) in QMIX. By this way, we can prove this Corollary in the same approach as for showing after the ReBorn process, QMIX satisfies the IGM principle. It is omitted for brevity. D
C Algorithm
The ReBorn algorithm is described in Algorithm 1.
Algorithm 1 ReBorn
Require:
dormant threshold a, over-active threshold 3, reborn interval ''' 1: Initialize parameters 0 of the mixing network
2: Initialize parameters o of the agent network
3: Initialize replay buffer D
4: for {1..... m episodes] do
5: Start new episode;
6: while episode is not _ot_end do
7: Get the Agent action
8:
Execute a, obtain global reward r and the next state s
9:
Update replay buffer 'D
10:
Sample batch from replay buffer D Loss(0, Φ) (Q(s, a; 0,Φ) ys.a)²
11:
12:
Update and @ and by Loss
13:
if mod T == 0 then
14:
Sample x from replay buffer
15:
Calculate the of of each neuron in mixing network
16:
Get dormant neurons dorm, which sf <a Get over-active neurons over which sf > for each over! do
17:
18:
19:
Randomly select K dorm, that have not been selected before over assign weights to dorm end for
20:
21:
Reinitialize input weights of unassigned dorm 23: Set output weights of unassigned dorm to
22:
24: end if
25:
end while
26:
end for
20

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D Experimental Details
D.1 Experimental Setup
We select 4 classical algorithms (QMIX, QPLEX, DMIX, RMIX) with different types to test the generality of ReBorn. QMIX and QPLEX are two well-known value-based MARL value factorization algorithms. DMIX is a distributional MARL value factorization algorithm, while RMIX is risk- sensitive MARL value factorization algorithm. These four algorithms cover multiple directions in the field of value factorization, demonstrating the strong applicability of ReBorn. Below is the brief descriptions of these algorithms.
Table I: Baseline value factorization algorithms
Algorithms
Brief Description
QMIX [2]
Learns a mixer of individual utilities with monotonic constraints
QPLEX* [10]
Learns mixer of advantage functions and state value functions
DMIX [11]
Integrates distributional RL with QMIX
RMIX [12]
Integrates risk-sensitive RL with QMIX
We implement these algorithms based on their open-source repositories to carry out performance analyses, with hyperparameters consistent with those in PyMARI. Our methods are implemented within the PyMARL framework, and each is evaluated using 5 random seeds, with 95% confidence intervals. Specific hyperparameters of different algorithms are listed in Table 2. We conduct experiments cluster equipped with multiple NVIDIA GeForce RTX 3090 GPUs.
Table Hyperparameter different value factorization algorithms
Hyperparameter
QMIX epsilon greedy
QPLEX
DMIX epsilon greedy
RMIX
Action Selector
epsilon greedy
epsilon greedy
Batch Size
32
32
32
32
Buffer Size
5000
5000
5000
5000
Learning Rate
0.0005
0.0005
0.0005
0.0005
Optimizer
RMSprop episode runner
RMSprop episode runner
RMSprop episode runner
Adam
Runner
episode runner
Mixing Embed Dimension
64
64
64
64
Hypernet Embed Dimension
64
64
64
64
RNN Hidden Dim
64
64
64
64
Target Update Interval
200
200
200
200
Discouunt Factor (7)
0.99
0.99
0.99
0.99
Dormant Threshold
0.1
0.1
0.1
0.1
3 Over-active Threshold
3
Execution Interval(Step)
200000
200000
200000
200000
In deep RI., ReDo [4] and ReSet [5] are two common mechanisms for improving network's perfor- mance through neuron processing. The specific introductions are as follows.
ReDo. ReDo periodically detects dormant neurons within the neural network and resets the input and output weights of these dormant neurons. The input weights are initialized using the Xavier method, while the output weights are set to zero.
ReSet. ReSet periodically resets the parameters of the neural network's 's final layer using Xavier initialization.
hhtps///gtttuhcco/////7//00
"narps//gihuuhcom///sooooddd
21

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Below is a brief introduction to all the methods used in the experimental section.
Table 3: Methods in Experimental Section
Algorithms
Brief Description
algorithm- ReBorn
Apply ReBorn the Mixing Network of algorithm
algorithm ReDo
Apply ReDo to the Whole Networks of algorithm Apply ReSet to the Whole Networks of algorithm Apply mechanism to the Mixing Network of al algorithm Apply mechanism the Mixing Network of algorithm
algorithm ReSet
algorithu- ReBorn (mechanism)
algorithm- mechanism with KI algorithm- mechanism w/o KI
Apply mechanism to the Whole Networks of algorithm
In Table 3, algorithm QMIX, QPLEX, DMIX, RMIX. mechanism is ReBorn, ReDo, ReSet. Specific hyperparameters of different mechanisms are listed as follows.
D.2 Environment
D.2.1 Predator-prey
Predator-prey simulates a grid world where multiple agents collaborate to capture preys dispersed throughout the map. At each time step, each agent choose of view. A prey is considered captured successfully only when at least two agents around it execute the capture action simultancously. Each successful capture brings team reward of +10, with the goal being to accumulate as much team reward as possible within a limited number of time steps. We develop three distinet environmental configurations small, middle, large, each featuring different numbers of agents and preys, as well as varying map sizes. Table 4 shows different environmental configurations of Predator-prey in detail.
capture within its local field
Game Rules
Agent Movement: Agents can move in four directions or stay in place. Movement is restricted by the presence of other agents or preys.
Observation and Decision Making: Each agent observes 3x3 grid centered around itself, receiving information about nearby agents and preys. Decisions are based on this local observation.
To capture prey, at least two agents must be adjacent to it and must choose the capture action at the same time. Successful capture relies strategic positioning and synchronized actions among agents.
Capture Mechanism:
Rewards and Penalties: Agents receive positive reward for each prey captured through cooperative action, while individual movement incurs a slight negative time penalty -0.1 to encourage efficiency.
Episode Termination: An episode terminates if all preys are captured or after predefined number of steps, providing fixed time frame for agents to maximize their collective reward.
Configuration
Number of Predators
Number of Preys
Map Size
Reward for Capture
Small
12
20 20
+10
Middle
12
24
30 x 30
+10
Large
18
36
40 40
+10
Table 4: Comparison of Predator-prey Configurations
22

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D.2.2 StarCraft II Multi-Agent Challenges (SMAC)
The StarCraft Multi-Agent Challenge (SMAC) [13] is popular benchmark used extensively in the domain of multi-agent reinforcement learning. Built on the StarCraft II game engine, SMAC specializes in micromanagement scenarios where each agent is controlled by independent agent that must make decisions based on local observations. MARL algorithms coordinate team of agents to engage in combat against opposing team managed by the game's built-in Al. The performance of these algorithms is quantitatively evaluated by the test win rate or the test return of the gameplay.
Name
Difficulty
Ally Units
Enemy Units
3s vs 5z
Hard
3 Stalkers
Zealots
2c vs vs_64zg
Hard
Colossi
64 Zerglings
1 Medivac, 2 Marauders
1 Mcdivac, 3 Marauders
MMM2
Super Hard
Marines
Marines
27m_ v____s_00mm 3s5z_vs vs_3s66
Super Hard
27 Marines
30 Marines
Super Hard
3 Stalkers & 5 Zealots
3 Stalkers & 6 Zealots
corridor
Super hard
Zealots
24 Zerglings
Table 5: Overview SMAC scenarios used in the experiment.
Table 5 depicts the overview of SMAC scenarios used in the experiment.
D.2.3 SMAC₂2
SMACv2 [14] addresses several critical limitations of SMAC, including the lack of stochasticity and partial observability. Unlike SMAC, SMACv2 features units that are randomly generated and positioned, enhancing stochasticity and significantly increasing the complexity of the scenarios.
Scenario Name
Number of Allies
Number of Enemies
Unit Types
10gen_zerg 10gen_terran
10
11
Zergling, Hydralisk, Baneling Marine, Marauder, Medivac Stalker, Zealot, Colossus
10
greeo_s
10
11
Table 6: Overview of SMACv2 scenarios used in the experiment.
Table 6 depicts the overview of SMACv2 scenarios used in the experiment.
D.3 Dormant neurons limit the expressive power of Mixing networks
To analyze the impact of the dormant ratio on the expressive power of the mixing network, we consider an illustrative example that requires mixing individual utilities of three agents. For this purpose, we design a simple 2-layer MLP network. The input layer, with a size of 3, receives individual utilities [Q:]²_ The hidden layer contains 4 neurons and uses Rel.U the activation function. The output layer, with a size of 1, produces Qtot- The objective is to fit the mixing function Qior 0.5 Q + Q + 1.5 Q3, Qi ~ N (0, 1).
We control the dormant ratio by varying the number of dormant neurons in the hidden layer. Figure (a) illustrates the expressive performance of networks with different dormant ratios. Number = indicates that there are dormant neurons in the hidden layer. We use Mean Squared Error as the loss function. According to the results, an increase in the dormant ratio will lead to reduced expressive power of the mixing network.
D.4 Experimental Results
D.4.1 ReBorn can improve the performance of various value f'actorization algorithms
23

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Predator-Prey & SMAC
predeton
GMIK
200
GMIK
GMIK
100
OPLEX
OPLEX
GPLEX
GPLECR
150
OPLEC
INIX
Rature
300
DHSS:
Raturn
150
40
20
50
0
200K
400K
Environmental Stops
Envirommental Stapo
Environmental Stups
predator-prey small
predatonprey medium
predston-prey
GPENK
50-
OPEEK
OME
Domant Porcent (S6)
Durmant Porcent [S]
OPLAX
GrLEx-tabore
ReBre
GeLEx
SHEX
40
To
30
1.
20
10-
200K
400K
GOOK
BOOK
1M
200K
400K
GOOK
BOOK
IM
200K
400K
GOOK
BOOK
IM
Envmentaaa Steps
Envl ronmentallllaaaoooomo Steps
Environmental Steps
100
5z
100
MMM22
1.0
27m_vs_300 v5_3_
80-
QHIIK Rollarm
80-
QUESK
OFLEX
ONCI
DRIK
TIME Win RutEe
DHIK
Test Win Rate
60
60
DAEE Belter
HNIS-kaboor
Text
40
40
20
20
400K
B00K
3.2M 3.0M
ZH
400K
BOOK
3,2M
3.4M
2H
0.0
095K
BOOX
1.6M
2M
Envirro momnnal Stapa
Envirommental Stape
kevirommental Steps
vs 5z
MMM2
27m, vs_ 20_30m
OPER-ieaaar
Dermant Percent [%
Wormannt Percent [N3
OPLLY
OPLEK
SPEE
SPEE
BMIX
48-
BMIX
20-
4D0K
3,2M
3.40
400K
BOOK
1.2M
2.
400%
B0
1.2M
3.00
2M
300
2c_ys_ 2c_vs_644u
100
3:52 vS 3o62
corridor
QMIBK Reborr
OPLEX-R-B om
80
QULEX
80-
QPLIXC
Test Win Rate
BHIK
Test Winh Rate
DMIK
60
60
RMIX
Text
40
40
1M0
2.4M
4.0M
0.801
2.4M
4.0M
Invironmental
Environmental
2c_vs_ 642g
3:52
3o62
corridor
Orttkllbbom
70
70
40
QPIL
Dommant Porcant (N6)
60
OPLEX
beee
Dormant Parcart
50
40-
Dormant
30
SPESS Sulter
15 8
400K
B0/K
1.2M
I.6
ZM
0.BM
1.6M
2.4M
3.2H
4.0%
0.8M
1.6M
2.4M
3.2M
4.0%
ronmental Stepss
Environmental Stops
Environmental Stops
Figure 3: ReBorn can improve the performance of various value factorization algorithms in Predator- Prey and SMAC.
24

---

SMACv2
10gen_perg
10gen_terran
10gen,
0.4
GPL Lomekom
Test Win Rato
0.5-
GrEadetion
GPL (PL.kntaiorm
Git
0.4-
9.2
Test
0.2
Tast
0.15
0.1
0.3
0.05
0.10
1.00
2,0M
3,4H
6.0H
5.0M
1.0M
2
3,0M
4.0%
5.0M
1,49
2,0H
3,00
4.0M
5.0M
10geri,zer zerg
Leren_terran
10gen prrotoss
Dormant Percont [50
DME Aethre
Durmant Parcent [%6]
76
BMI Sebore
00-
OPLESc tedlooe
OPLEk-ke kebor
OPLEN herbrr
V
M Wmm Mam
3,601
3,606
3,0M
4.00
5.0M
3.0M
2,0H4
3:0M
3.0M
5.IM
1,00
2.0M
2,00
4.000
5.0M
Envtr nnvirommettl Stops
Gnvtronmmnntal Strps
Enstronmenta Stops
Figure 4: ReBorn can improve the performance various value factorization algorithms in SMACv2.
D.4.2 Compare neuron activation values with different methods
QMIX
QMIX ReB orn
QMIX-Redo
50
50
50
40
40
40
Poreent [%)
30
Parcant
20
20
25
10
15
20
25
10
1.5
20
25
Score Ranking
Score Ra ricing
Score Ranicing
Figure 5: The Normalized Activation Score percentage ranking for top-25 over-active neurons in 27m_vs___ 27m_vs_30m.
D.4.3 ReBorn better than other methods that satisfy the KI principle
MMM2
MMM2
MMM2
outt-lation
OPLES- PLLE--aaom (isceeT) 0.8
0.8
gwilokation Seaded
Scneeaaam (RASH)) nadol
5N00-aators Smuning
OPLL5- Radema (mruiing)
Test Win Rate
0.8
3.6
Tost Win
6.76
1.0
1.76
2
0.25
3,50
1J5 1.50
0.25
1.55
3.50
175
Envireemental Steps
Eevireemental Stops
Bew onmeental Stops
MMM2
MMH2
MMM2
(miokeakers
GPLL3 kabtra
(habet)
Dormant Ratio E55
a (Radect
Dormant Ratio 196]
42-
2013--te (Rader)
(Pronlng!
gPiE Sothore (Trerinss)
Dormant
30
20
InviotmentalSoo
Enstronmental Stops
Enstranmental Steps
Figure 6: Comparison with other methods that satisfy the KI principle.
25

---

D.4.4 ReBorn is superior to other RL parameter perturbing methods
pradalor-gtey smal
60
Dormunt Purrant 153
Overr Parmund
00
proceate
becomese
C2-
-0
20
20
0.4M
6.6M
Environmennt Stopa
Environmental Steps
Enverommental Stepo
predetor-gheey
predaton-pre pray medium
predstooppeey
2.00
CAP7
Ovemanthe Sarn Parcost (10)
100
Dormant Purcent PM
Rebum
WW S
20
Covirsmmeennall Steps
Environtmental Steps
Ennfronmental Steps
VS. 642g
2c.v5 642g
2c.vo 640gg
60-
2013
Sam Parcennt 153
Dereunt Partann 113
Win 2184
20
Enstronmental Sttpps
Invirommental Steps
Invirenmentall Stepo
10gann 2arg
-3
Overoeight Sum Porcont [N]
Certrnebure
Purcent [N]
Text Win 837
Durmant Purcent
2 22
Wymmm
0.8
3.38
2.8M
4.OM
5.5M
1.2M
2,3H
4,06
1
2,00
2:4m
4.50
$.00
Figure Comparison with Related Methods.
D.4.5 ReBorn can improve the performance of ResQ
predatton pray small
predstan pray medium
precitoor pray harge
35
Bm
70
50
25
50
Neture
40
30
20
20
10
200K
4D0K
ROOK
ROOK
1M
200K
6OOK
GOOK
BOOK
1M
200K
4010
BOOK
BOOK
MMN2
277m
dualer
NerO-littire
80
80
Text With Rata
Teet Mike Batts
Taet Win
60-
800K
1.3M
4B0K
400K
BOOK
1.3M
1.4M
Figure 8: ReBor improve the performance of ResQ.
26

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E
Discussion
E.1 Societal impact
Our research primarily concentrates on the technical and theoretical aspects of multi-agent reinforce- ment learning, aiming enhance the performance of these agents across a variety of tasks. While we do not foresce any direct negative consequences arising from our research, we are committed maintaining an open dialogue. We highly appreciate and value constructive feedback from the community to ensure our work's contributions are beneficial and ethically sound.
E.2 Limitations and future work
Although our proposed simple recycling method has achieved good results across various algorithms, there is still room for further improvement. We have defined dormant neurons and over-active neurons in straightforward manner. However, their identification should not be limited normalized activation values. More precise identification could be achieved by considering additional factors such as update gradients and output weights. We studied the phenomenon of dormant neurons in discrete multi-agen environments. Future work should explore whether our method can be extended to continuous environments. Regarding different thresholds and recycling periods, setting threshold too high or recycling too frequently can disrupt the network's normal learning process. Conversely, low thresholds and infrequent recycling can reduce the effectiveness of the recycling process. Therefore, developing adaptive thresholds and recycling mechanisms will be key focus of future work.
27

---

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Reviews:
[{"review_id": "ndLbhfwIq3", "reviewer": "Reviewer_KvCg", "summary": "This work proposes a new method for resetting dormant neurons in multi-agent RL settings. It replicates previous observations on unit dormancy in deep RL in the multi-agent regime, finding that inactive neurons are correlated with reduced ability to improve performance. The proposed method differs from ReDO, which resets dormant neurons' incoming weights to that of a random initialization, by distributing the weights of highly active \"overweight\" neurons among those identified as dormant during reset steps so as to avoid changing the output of the network.", "questions": "1. Can the authors comment on how they tuned ReDo and give some insight into why it seems to not have any effect on the number of dormant units in Figure 6?\n2. Please address my questions regarding theorems 1 and 2. Are there instances where even with a monotone mixing network, a parameter-perturbing method will fail to satisfy IGM?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- The paper identifies additional factors, beyond those present in single-agent deep RL, which impact the dormancy rate in a neural network, in particular showing that in QMIX and QPLEX algorithms, the *mixing* network is most vulnerable to dormancy. It also shows that increasing the number of agents, which presumably also increases the degree of nonstationarity in the problem, also increases the number of dormant units.\n- The knowledge invariant principle, while not entirely novel (see, e.g. the approach of Nikishin et al. which is precisely motivated by the desire to avoid changing the network's outputs after a reset), is clearly relevant for multi-agent settings, where changing many agents' policies at once introduces greater instability than the (potentially beneficial) exploratory interpretation of parameter perturbations in single-agent algorithms.\n- The observation that the accumulation of dormant neurons is accompanied by accumulation of 'overweight' neurons is sensible and points to an interesting alternative approach to maintaining plasticity: rather than focusing on resetting units which are not receiving gradients, it may be beneficial to re-distribute utility across the units in a layer. The proposed solution makes a lot of sense from this framing.", "weaknesses": "I have two main concerns with this paper which prevent me from confidently recommending its acceptance: the first is the validity of the theoretical results, and the second is signficance of the empirical results. I list more minor concerns later, which may benefit the paper but which will not affect my decision.\n\n- Major: while it is true that a parameter perturbation which changes the rankings of global actions can cause a function which previously satisfied the IGM principle to violate it, such a perturbation would also violate the assumption on the monotonicity of the global value w.r.t. agent utilities which is baked into methods like QMIX. Indeed, the proof of Theorem 1 does not provide a concrete example of a perturbation to the network parameters which violate the KI principle and violate the IGM principle, as such an example which satisfies [B.5] would require non-monotonicity of the mixing function. Since the algorithms studied in the paper do use monotone mixing functions, Theorem 1 does not seem relevant. \n\n- Major: I could not find anything in the proof of Theorem 2 which depends on the particular form of the ReBorn update. Instead it seems that Theorem 2 depends on the monotonicity of the QMIX mixing network (b.10), rather than any particular property of the perturbation to $\\hat{\\theta}$. This further reinforces the above concern that the proposed failing of parameter perturbation methods in the general case of Theorem 1 is in fact only a product of relaxing the assumption on the monotonicity of the mixing network.\n    \n- Major: In many instances, using ReDo appears to hurt performance compared to naive QMIX, despite reducing the dormancy percentage. Further, it appears that in some environments, applying ReDo does not change the number of dormant neurons. This causes me to wonder if the ReDo method has been appropriately tuned for the domains it is being applied to. \n\n- Minor: to me, a better name for the so-called 'overweight' neurons would be 'over-active', since it's not clear that the cause of their disproportionate activity is due to the magnitude of their weights or to the alignment of their incoming weights with the previous layer's features. This also makes it more clear their relationship with dormant neurons.\n\n- Minor: there are a few grammatical issues in the paper that would benefit from review. For example, the word 'albeit' is typically used at the start of a phrase, and not a clause as is often done in the paper, where it can generally be replaced with 'although'.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your valuable comments, we will improve our work based on your suggestions. We address your concerns as follows. \n\n>The knowledge invariant principle, while not entirely novel (see, e.g., the approach of Nikishin et al., which is precisely motivated by the desire to avoid changing the network's outputs after a reset)\n\nThe approach of Nikishin et al. is the Reset method used for comparison. It aims to preserve the experience within the replay buffer and the un-reset layers. However, it does not try to avoid changing the output of network. \n\n\n>Major 1: while it is true that a parameter perturbation which changes the rankings of global actions, ... not provide a concrete example of a perturbation to the network parameters which violate the KI principle and violate the IGM principle, ... require non-monotonicity  ... Theorem 1 does not seem relevant.\n\n\nHere, we present an example that violates the KI and IGM principles. Assuming we have trained a QMIX-like value factorization method $f$ that implements $Q_{tot}=\\sum_{i=1}^Nk_i\\times Q_i$, where $k_i \\ge 0$. It is implemented by a two layer-neural network parameterized by $k_i$. In the last layer, there is only one neuron (output neuron). For each neuron $i$ in the first layer, its input is individual utility $Q_i$, and the weight connected to the output neuron is $k_i \\ge 0$. The activation function of the output neuron is an identity function. Clearly, $f$ is a monotonic increasing network, satisfying the IGM principle. If a parameter perturbing method changes all the weight $k_i$ to be smaller than zero, then $f$ does not satisfy the IGM principle, and the KI principle is violated.\n\nWe agree with the reviewer that *[B.5] would require non-monotonicity of the mixing function*. We have shown that a perturbing method could change a mixing function from monotonic increasing to monotonic decreasing. \n\n\nBesides monotonic increasing mixing functions (e.g., QMIX and RMIX), ReBorn supports both DMIX and QPLEX. DMIX has a **non-monotonic increasing** mixer for value distributions and QPLEX can model **non-monotonic** relationships. We implement ReBorn on the hypernet for QMIX, on the QMIX part of DMIX, and advantage part of QPLEX to ensure that we do not change their functional relationships. We will make this clearer. \n\n\n>Questions 2: Are there instances where even with a monotone mixing network, a parameter-perturbing method will fail to satisfy IGM?\n\nIf a parameter-perturbing method changes the functional property of the mixing network, it will fail to satisfy IGM. If the functional property is preserved, it will not fail to satisfy IGM. \n\n>Major 2: I could not find anything in the proof of Theorem 2 which depends on the particular form of the ReBorn update. ... only a product of relaxing the assumption on the monotonicity of the mixing network.\n\nThe proof of Theorem 2 does depend on the implementation of ReBorn on QMIX. We will make it clearer in the paper. We apply ReBorn to the neurons of hypernetworks used in QMIX. Hypernetworks are used to generate non-negative weights of the mixing network. A hypernetwork takes $\\tau$ as input and outputs the weights of a layer of the QMIX mixing network. A hypernetwork consists of a single linear layer, followed by an activation function, which ensures that the mixing weights are non-negative after ReBorn. \n\nAfter applying ReBorn, the monotonic increasing property of QMIX does not change. We do not perturb the agent network, as we find that the dormant ratio is low in the agent network. In Sec 6.4.1, we apply ReBorn both to the mixing network and the agent network. This leads to a violation of the KI principle. As shown in Fig. 7, depicted as ReBorn w/o KI, applying it only to the mixing network leads to better performance.\n\n\n\n>Major 3: In many instances, using ReDo appears to hurt performance compared to naive QMIX, despite reducing the dormancy percentage ... applying ReDo does not ... dormant neurons ...  if the ReDo method has been appropriately tuned ...\n\n>Question 1: Can the authors comment on how they tuned ReDo and give some insight into why it seems to not have any effect on the number of dormant units in Figure 6?\n\n**For Figure 6, ReDo can reduce dormant ratios.** The middle of Figure 6 (b, e) depicts the dormant ratios for different methods. The results for ReDo (QMIX-ReDo) are depicted as a blue curve, showing that it can reduce the dormant ratio for the QMIX algorithm (depicted as a green curve).\n\nReDo can **reduce the dormant ratio** for all the cases shown in this work. Please refer to all the figures (Fig. 6(b, f), Appendix Fig. 5 middle, Appendix Fig. 7 bottom row) that depict the dormant ratio for Redo. Regarding performance, ReDo hurts the performance of QMIX on MMM2, but it increases its performance for 27m-vs-30m, which has a higher dormant ratio than MMM2.\n\nFor a fair comparison, we performed parameter turning on ReDo for 2 SMAC environments (MMM2 and 5m vs 6m) based on ReDo's default parameters, which perform well for 47 single agent environments. We explored two parameter re-initialization methods (Xavier and Kaiming) and two dormant thresholds (0.025 and 0.1). The results are depicted in Figure 5 of the response PDF. Different configurations perform similarly. In the end, we choose the default parameters of ReDo.\n\nAs we discussed in Sec. 4, the presence of over-active neurons impacts the existence of dormant neurons. Therefore, ReDo's low efficiency in reducing dormant neurons might be due to its inability to handle over-active neurons. This is a factor contributing to the effectiveness of ReBorn, which addresses both dormant and over-active neurons.\n\n\n>Minor: a better name for the so-called 'overweight' neurons would be 'over-active'.\n\nWe will change the word \"overweight\" to \"over-active\".\n\n>Minor: there are a few grammatical issues in the paper ...\n\nWe will change the 'albeit' in a clause to 'although'. We will check the paper carefully to improve its overall quality.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MEmztHuCur", "reviewer": "Reviewer_hhg9", "summary": "The paper explores the dormant neuron phenomenon, an active research topic in RL, in the context of MARL value factorization. It describes how dormant neurons manifest mostly in the mixing network, details their effect on performance and connects them to an opposite but correlated phenomenon, overweight neurons. The paper proposes a method to counter the dormant neuron phenomenon by redistributing the weights of overweight neurons among dormant neurons. In MARL value factorization, the proposed method is demonstrated to outperform general RL methods for dealing with dormant neurons.", "questions": "Suggestion:\nEvaluation on more environments could strengthen the paper's claim of superiority over other methods. However, I find the current evaluation experiments sufficient to make that claim.", "limitations": "Limitation are not discussed in the main section, only in the appendix.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 4, "strengths": "The paper text is easy to read and understand, and the problem the paper investigates is important and topical. Evaluation shows that, on two environments, the proposed method (ReBorn) is clearly better than general RL methods used to solve this problem. The discussion on the existence of overweight neurons is also significant for the field outside of the specific problem setting.", "weaknesses": "The figures and figure captions are often hard to understand. Some plots are hard to interpret solely from the plot and caption text, e.g. to understand what 'Number' means in Figure 2a, one must look for explanations in the text body. Another example is Figure 4, which the reader is referred to for a depiction of the main method of the paper, but contains a total of 8 words in legends and caption combined. In contrast, the figures themselves are visually clear to understand.\nThis problem can be easily fixed by expanding some of the captions and using more expressive wording in figure legends and titles. I am willing to raise my score if the figures are presented in a more readable manner.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer's time and effort to review this paper. We will improve the quality of the figures and add more experimental results to our work. \n\n>The figures and figure captions are often hard to understand.\n\nThanks for your suggestion on Figure 2a and Figure 4. We have improved the presentation of these two figures. They are presented in Figures 1 and 3 in the response PDF. Additionally, we have improved the readability of Figure 2(b), Figure 2(c), Figure 3(a-c), Figure 5(a-c), Figure 6 (a-c), Figure 7 (a-c), and Figure 8 (a-c) as follows.\n\nFor Figure 2a, we have changed its vertical label from \"Loss\" to \"MSE Loss for a Mixing Network\" and changed its legend from \"Number\" to \"Number of Dormant Neurons\". Further, we have also updated the caption of the Figure from \"Dormant neurons hurt mixing network expressivity\" to \"The MSE Loss for fitting a simple Mixing Network increases with an increasing number of Dormant Neurons. It indicates that dormant neurons hurt mixing network expressivity.\" \n\nFor Figure 4, we have updated the caption of the Figure from \"ReBorn Method\" to \"The procedure of ReBorn neurons. The weights of overweight neurons are distributed to $M$ randomly picked dormant neurons. $w_x^{in}$, $w_x^{out}$, and $b$ are the input weights, output weight, and bias for an overweight neuron. $w_i^{in}$, $w_i^{out}$ and $b_i$ are the input weights, output weight, and bias for dormant neuron $i$. After Reborn, the input weights and bias for a dormant neuron $i$ becomes $\\beta_i\\alpha_iw_x^{in}$, and $\\beta_ib_x$, where $0\\leq \\alpha_i \\leq 1, and \\sum_{i=0}^M\\alpha_i=1$, $\\beta_i$ is a random number among [0.5, 1.5].\"\n\n\nFor Figure 2(b), we have relocated the legends from the top-left to the bottom-right and changed the legend from \"Interval\" to \"Update Interval.\"\n\nFor Figure 2(c), we have changed its vertical label from \"Percent\" to \"NAS Percentage\", and changed its horizontal label from \"Score ranking\" to \"NAS ranking for top-25 neurons\". We have updated its caption from \"Overweight neurons in the QMIX mixing network\" to \"The Normalized Activation Score (NAS) percentage ranking for top-25 overweight neurons in the QMIX mixing network.\"\n\nFor Figure 3, we have updated the caption to \"Overweight neurons in QMIX mixing networks: (a) The percentage contribution of the number of dormant neurons (depicted as Dormant), the number of overweight neurons (depicted as Overweight-Number), the sum of NAS (depicted as Overweight-Sum) for overweight neurons over time. (b) Overlap coefficients for Dormant/Overweight neurons between the current iteration and previous iterations. (c) Percentage of dormant neurons that re-enter dormancy after ReDo within different time steps.\"\n\nFor Figure 5 (a-c), we have changed the caption from \"ReBorn can improve the performance of various value factorization algorithms\" to \"ReBorn can improve the performance of various value factorization algorithms: (a-b) the test win rate for the 3s5z\\_vs\\_3s6z and the MMM2 environments, (c) the return for predator-prey small environment, (d-e) the dormant percent for the the 3s5z\\_vs\\_3s6z, the MMM2, and the predator-prey small environment.\"\n\nFor Figure 6 (a-c), we have updated the caption from \"Comparison with other Parameter Perturbing Methods.\" to \"Comparison with other Parameter Perturbing Methods: (a-c) The test win rate, the dormant percentage and the percentage of the sum of normalized activation score (NAS) for the MMM2 environment. (d-f) The test win rate, the dormant percentage, and the percentage of the sum of NAS for the 27m\\_vs\\_30m environment.\"\n\nFor Figure 7 (a-c), we have changed the caption from \"Importance of satisfying the KI Principle\" to \"Importance of satisfying the KI Principle for (a) QMIX, (b) QPLEX, and (C) RMIX. A variant of ReBorn without satisfying the KI Principle is depicted as Reborn w/o KI.\"\n\nFor Figure 8 (a-c), we have changed the caption from \"Comparison with other methods that satisfy the KI principle\" to \"Comparison with other methods that satisfy the KI principle for (a) QMIX, (b) QPLEX, and (C) RMIX.\"\n\n\n>Evaluation on more environments could strengthen the paper's claim of superiority over other methods. However, I find the current evaluation experiments sufficient to make that claim.\n\nWe have evaluated ReBorn against ReDo and Reset in four more environments. The experimental results are shown in Figure 2 in the response PDF. The results demonstrate that ReBorn performs better than these two methods.\n\n\n>Limitation are not discussed in the main section, only in the appendix.\n\nThank you for your valuable suggestion. We will incorporate the discussion of the limitations in the main section.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IXfCmDqfXI", "reviewer": "Reviewer_JB6L", "summary": "This paper introduces a novel approach for tackling the plasticity loss and securing sample efficiency in multi agent reinforcement learning (MARL). Different from the methods in single agent RL, this paper also shows the importance of not violating the knowledge invariant (KI) principle in MARL. With careful analysis and extensive experiments, the proposed method, ReBorn, outperforms other baselines such as ReDo or Reset. The authors said that the baselines, ReDo and Reset, violate the KI principle, thus they are not appropriate to be applied to MARL setting. However, the proposed method, ReBorn, can satisfy the KI principle, leading to achieve remarkable performance.", "questions": "Already mentioned in the Weakness section.", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- One of the most important part in this paper, the KI principle, is well defined, and can represent the constraints in MARL setting. Due to the difficulty of satisfying this principle, applying previous methods is not straightforward. Nevertheless, the proposed method, ReBorn, does not violate this principle, and outperforms the baselines.\n\n- Scaling the input and output weights connected to overweight neurons is a novel approach for resolving the plasticity loss problem in Rl community. This paper pointed out the drawback of ReDo in MARL scenario, and proposed a novel approach.", "weaknesses": "- The motivation behind scaling the weights in ReBorn is not clear. Though scaling the weights to ease the strength of the overweight neurons is straightforward, is it only way to reduce the number of overweight neurons? I'm quite confusing why we should use the scaling approach in ReBorn.\n\n- I think there is a missing baseline [1]. This approach can also effectively resolve the plasticity loss problem, and achieves high sample efficiency in singe agent RL. Although this approach violates the KI principle, with high replay ratio, it may achieve good performance in MARL setting. Similar to [1], a recently proposed method [2]  show that resetting the network and training the network with high replay ratio in MARL setting can outperform the baselines. I know it is difficult to  compare ReBorn and the method in [2] at submission phase, but I wonder the method in [2] violating the KI principle is truly failed to overcome the plasticity loss.\n\n\n[1] D'Oro et. al. ,Sample-Efficient Reinforcement Learning by Breaking the Replay Ratio Barrier, ICLR, 2023\n\n[2] Yang et. al., Sample-Efficient Multiagent Reinforcement Learning with Reset Replay, ICML, 2024", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for your time and effort in reviewing this work. Your suggestions are valuable. We have compared our method with the two approaches mentioned by the reviewer. We address the concerns of the reviewer as follows. \n\n>The motivation behind scaling the weights in ReBorn is not clear. Though scaling the weights to ease the strength of the overweight neurons is straightforward, is it only way to reduce the number of overweight neurons? I'm quite confusing why we should use the scaling approach in ReBorn.\n\nWe agree with the reviewer that there could be multiple ways to ease the strength of the overweight neurons. We did explore another way of distributing the weights, which set $\\alpha_i$ = $1/(M+1)$, and $\\beta$ = 1. As shown in Figure 4 of the response PDF, such an average weight-sharing method performs inferior to ReBorn; its performance is weaker than the weight scaling approach. We would like to explore the method used by [2], which uses a Shrink & Perturb strategy.\n\n>I think there is a missing baseline [1]. This approach can also effectively resolve the plasticity loss problem and achieve high sample efficiency in single-agent RL. Although this approach violates the KI principle, with a high replay ratio, it may achieve good performance in the MARL setting. Similar to [1], a recently proposed method [2] shows that resetting the network and training the network with a high replay ratio in the MARL setting can outperform the baselines. I know it is difficult to compare ReBorn and the method in [2] at the submission phase, but I wonder if the method in [2] violating the KI principle truly failed to overcome the plasticity loss.\n\nWe will add the following discussion to the related work section. *[1] increases replay ratio and resets all the parameters of networks based on the number of updates. For MARL, [2] periodically resets the parameters of agent networks and uses data augmentation to further increase the replay ratio.* \n\n\nWe have evaluated the performance of [1] and [2] in four more environments. The results are depicted in Figure 2 of the response PDF. As shown in Figure 2, these two methods perform inferior to ReBorn. We will add these experimental results to our work. \n\nWe agree with the reviewer that the KI principle offers a perspective on the plasticity loss problem. There could be more perspectives on this issue.\n\nReferences\n\n[1] D'Oro et. al., Sample-Efficient Reinforcement Learning by Breaking the Replay Ratio Barrier, ICLR, 2023\n\n[2] Yang et al., Sample-Efficient Multiagent Reinforcement Learning with Reset Replay, ICML, 2024", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aVBSYv13L1", "reviewer": "Reviewer_WpXH", "summary": "The paper proposes ReBorn to reactivate dormant neurons in the mixing network of multiagent value-mixing algorithms. Specifically, ReBorn transfers the weights from overweight neurons to dormant neurons and this method ensures the learned action preferences are not changed after the ReBorn operation. Experiments on SMAC, predator-prey, and SMACv2 demonstrate the effectiveness of the proposed method.", "questions": "1.\tCould the authors compare the running time of the baselines and their ReBorn variants?\n2.\tCould ReBorn be applied to other MARL algorithms besides the multiagent value-mixing algorithms such as MAPPO and MADDPG?", "limitations": "NA.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\tThe idea is novel and the motivation is well-explained.\n2.\tExperiments on different scenarios show that ReBorn improves the performance of different baselines.\n3.\tThe authors theoretically prove that ReBorn would not affect the learned action preferences.", "weaknesses": "1.\tIt seems that the ReBorn operation is time-consuming as it needs to compute and manipulate each neuron in the network. If the network is large, this ReBorn operation may become infeasible.\n2.\tThe ReBorn is only validated in the multiagent value-mixing algorithms.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for valuable comments and suggestions, which can help improve the quality of our work. We address the concerns of the reviewer as follows. \n\n>Weakness 1: It seems that the ReBorn operation is time-consuming as it needs to compute and manipulate each neuron in the network. If the network is large, this ReBorn operation may become infeasible.\n\n>Question 1: Could the authors compare the running time of the baselines and their ReBorn variants?\n\n\nThe time complexity of the ReBorn operation increases **linearly** with the number of neurons. In the 27m_vs_30m environment, ReBorn is performed 19 times. The training time of the QMIX algorithm increases from 23.8 hours to 24.6 hours. By applying ReBorn, the win rate of QMIX increases from 39% to 51%. The performance benefit (51/39=1.31) bring by ReBorn **outweighs its computational cost** (24.6/23.8=1.03). \n\nWe have compared the running time of ReDo and Reset with ReBorn, as well as with two new baselines, SR[1] and MARR[2]. The wall clock time is depicted in the table below. As shown in the table, ReDo, Reset, and ReBorn take almost the same amount of time. Additionally, SR[1] and MARR[2] take more time than ReBorn.\n\n|Map/QMIX|Baseline|ReSet|Redo|ReBorn|SR|MARR|\n|--|--|--|--|--|--|--|\n|3s_vs_5z(2M)|10.2|10.4|10.6|10.9|15.3|11.3|\n|MMM2(2M)|12.3|12.4|13.1|12.9|17.4|13.6|\n|27m_vs_30m(2M)|23.8|24.2|24.7|24.6|29.3|25.3|\n|2c_vs_64zg(2M)|18.3|18.7|19.2|19.3|24.2|19.6|\n|stag_hunt_s(1M)|3.9|3.9|4.2|4.1|6.3|4.8|\n|stag_hunt_m(1M)|7.8|8.0|8.5|8.3|10.3|8.9|\n|stag_hunt_l(1M)|12.7|13.1|13.4|13.6|18.6|14.2|\n\n\nWe agree with the reviewer that there could be better ways to reduce the computational overhead of ReBorn. For example, ReBorn could be applied only to a few layers of a neural network rather than the whole neural network. We plan to explore such alternatives in the future.\n\n\n\n> The ReBorn is only validated in the multi-agent value-mixing algorithms.\n\nAs it is written in the title of our submission: \"The Dormant Neuron Phenomenon in Multi-Agent Reinforcement Learning Value Factorization\", we focus on the value factorization (value mixing) algorithms.\n\nWe have applied ReBorn to the critic networks of MADDPG and MAPPO, which are not multi-agent value-mixing algorithms. The experimental results are shown in Figure 6 of the response PDF. As it is shown, ReBorn can reduce the dormant ratio of the critic networks of MADDPG  and MAPPO, and improve their performance as well. In future work, ReBorn is expected to be applied in other multi-agent fields.\n\n\n\nReferences\n\n[1] D'Oro et. al., Sample-Efficient Reinforcement Learning by Breaking the Replay Ratio Barrier, ICLR, 2023\n\n[2] Yang et al., Sample-Efficient Multi-agent Reinforcement Learning with Reset Replay, ICML, 2024", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We sincerely thank all reviewers for their insightful comments and valuable feedback. The reviewers acknowledge our work as novel (KvCg, JB6L, WpXH), important (hhg9, JB6L), significant outside the specific problem setting (hhg9), theoretical contributions (KvCg, JB6L, WpXH), good empirical results (hhg9, JB6L, WpXH). We will incorporate the suggestions and address the concerns in the new version of our work. We have conducted 56 additional experiments to address the comments, and their results are included in the response PDF. We describe the figures as follows.

**Experimental Results**

Fig. 1: We have re-designed the figure and improved its caption to make it more readable.

Fig. 2: Comparing ReBorn with more new baseline[1][2], Reset, ReDo. The experimental results show that ReBorn performs better than them.

Fig. 3: The MSE Loss for fitting a simple Mixing Network increases with an increasing number of Dormant Neurons. We have improved the legend and the caption of the figure to enhance readability.

Fig. 4: Replacing the weight scaling strategy used in ReBorn with an average weight sharing strategy. The experimental results indicate that the weight scaling strategy performs better.

Fig. 5: Hyper-parameter tuning for the ReDo method. These Hyper-parameters exhibit similar performance.

Fig. 6: Applying ReBorn to the critic network in MADDPG and MAPPO. ReBorn can reduce the dormant ratio of the critic network and improve its performance as well.

**References**

[1] D'Oro et al., Sample-Efficient Reinforcement Learning by Breaking the Replay Ratio Barrier, ICLR, 2023

[2] Yang et al., Sample-Efficient Multiagent Reinforcement Learning with Reset Replay, ICML, 2024

Meta-review: This paper studies the dormant neuron phenomenon in value factorization of multi-agent reinforcement learning (MARL) and finds that the dormant neurons negatively affect the learning process. With observing that dormant neurons correlate with the existence of overweight neurons, the authors propose ReBorn that transfers the weights from overweight neurons to dormant neurons with some theoretical analysis and promising empirical results. Overall, the story path of this paper is interesting, and the proposed method is important for making CTDE MARL methods more effective. During the rebuttal phase, the authors provided satisfactory responses to ease the concerns of the reviewers. As a result, I recommend acceptance of this paper.
Decision: Accept (poster)