Title: Scalable Constrained Policy Optimization for Safe Multi-agent Reinforcement Learning
Venue: NeurIPS 2024
Accepted: yes
Keywords: Multi-agent reinforcement learning

policy optimization

safe learning

scalable method
Abstract: A challenging problem in seeking to bring multi-agent reinforcement learning (MARL) techniques into real-world applications, such as autonomous driving and drone swarms, is how to control multiple agents safely and cooperatively to accomplish tasks. Most existing safe MARL methods learn the centralized value function by introducing a global state to guide safety cooperation. However, the global coupling arising from agents’ safety constraints and the exponential growth of the state-action space size limit their applicability in instant communication or computing resource-constrained systems and larger multi-agent systems. In this paper, we develop a novel scalable and theoretically-justified multi-agent constrained policy optimization method. This method utilizes the rigorous bounds of the trust region method and the bounds of the truncated advantage function to provide a new local policy optimization objective for each agent. Also, we prove that the safety constraints and the joint policy improvement can be met when each agent adopts a sequential update scheme to optimize a $\kappa$-hop policy. Then, we propose a practical algorithm called Scalable MAPPO-Lagrangian (Scal-MAPPO-L). The proposed method’s effectiveness is verified on a collection of benchmark tasks, and the results support our theory that decentralized training with local interactions can still improve reward performance and satisfy safe constraints.

Scalable Constrained Policy Optimization for Safe Multi-agent Reinforcement Learning

Abstract

A challenging problem in seeking to bring multi-agent reinforcement learning (MARL) techniques into real-world applications, such as autonomous driving and drone swarms, is how to control multiple agents safely and cooperatively to accomplish tasks. Most existing safe MARL methods learn the centralized value function by introducing a global state to guide safety cooperation. However, the global coupling arising from safety constraints and the exponential growth of the state-action space size limit their applicability in instant communication or computing resource-constrained systems and larger multi-agent systems. In this paper, we develop a novel scalable and theoretically-justified multi-agent constrained policy optimization method. This method integrates the rigorous bounds of the trust region method and the bounds of the truncated advantage function to provide a new local policy optimization objective for each agent. Also, we prove that the safety constraints and the joint policy improvement can be met when each agent adopts a sequential update scheme to optimize a κ -hop policy. Furthermore, we propose a practical algorithm called Scalable MAPPOLagrangian (Scal-MAPPO-L). The proposed method's effectiveness is verified on a collection of benchmark tasks, and the results support our theory that decentralized training with local interactions can still improve reward performance and satisfy safe constraints.

1 Introduction

With the advanced and rapid developments of reinforcement learning technology, many researchers have gradually shifted their focus from virtual simulation to real-world cyber-physical applications [1, 2]. In this process, safety challenges are inevitable, especially in multi-agent safety-critical scenarios, e.g., autonomous vehicle navigation [3], power grids [4], and drone swarms [5], in which agents perform complex cooperative tasks while adhering to a variety of local and system-wide limitations or constraints. These constraints can be derived from domain-specific knowledge and are intended to prevent damage to people or other environmental elements, such as equipment and infrastructure, or to prevent the inability to accomplish specific tasks or objectives. Take multi-robot control as an example. Each running robot must not take certain actions or not visit certain states, which may imply unsafe for itself, its collaborators, or the infrastructure of its environment [6]. These widespread potential dangers exacerbate the difficulty of safety decision-making when applying MARL. Consequently, it is necessary to research the safe decision-making problem in MARL to ensure that agents can work together safely and cooperatively to accomplish tasks.

There are two main approaches concerning safe MARL techniques in the existing literature. The first type is shielded-based reactive methods [7, 8], which combine environmental dynamics and safety specification constraints to predict whether the actions chosen by agents will violate cost constraints. Nevertheless, due to the reliance on precise modeling knowledge, these methods may lead to poor performance when the accurate state transition model is unavailable. The second type formulates the safe MARL problem as a constrained Markov game, which requires agents to solve a constrained optimization problem, i.e., maximize total reward while avoiding violating cost constraints. To mention a few, several safe MARL variants, such as CMIX [9] and MAPPO-L [10], have been proposed, which learn the centralized value function to overcome policy conflicts caused by the partially observable and non-stationarity nature of the environment faced by each agent. Unfortunately, the global coupling arising from agents' safety constraints and the exponential growth of the state-action space size make the usability of these algorithms in instant communication or computing resource-constrained systems and the scalability in larger multi-agent systems become a bottleneck, limiting their applicability.

A promising approach for avoiding these shortcomings, which has received attention in recent years, is to exploit networked application-specific structures. For example, Safe Dec-PG [11] employs a primal-dual framework to find the saddle point between maximizing decoupled rewards and minimizing costs under a consensus network. However, it is worth noting that this approach still assumes each agent can access the global state and requires that the actions of all neighboring agents on the network be available. Recent research [12] proposes a scalable safe MARL approach based on the spatial decay assumption of the environment dynamics, which updates the policies of agents by the truncated gradient estimators depending on the local states and actions of the κ -hop neighboring agents. However, due to the dependence on the actions and states of its neighbors, this method necessarily involves joint training in a local area, which is still plagued by non-stationary issues. Motivated by the urgent desire for scalable learning in practical applications and the fact that meeting both safety constraints and joint policy improvement is challenging for most methods, we investigate a novel scalable safe MARL with theoretical analysis, practical algorithm, and simulation verification.

Specifically, we focus on decentralized learning settings without global observability, where each agent can only access the local state information of itself and its neighbors. Our main contributions are summarized as follows.

We develop a novel scalable multi-agent constrained policy optimization method that eliminates dependence on the global state and other agent actions during each agent's training. Furthermore, we parameterize each agent's policy and propose a practical algorithm called Scalable MAPPO-Lagrangian (Scal-MAPPO-L).

We quantify the maximum information loss regarding the advantage truncation based on two assumptions about the transition dynamics and policies. Then, each agent's new local policy optimization objective is provided by integrating the rigorous bounds of the trust region method and the bounds of the truncated advantage function. In addition, we prove that the safety constraints and the joint policy improvement can be guaranteed when updating the local policy with a sequential update scheme.

Experimentally, we provide the results on several safe MARL tasks to evaluate the effectiveness of our proposed method and the sensitivity for the parameter κ . The results support our theory that decentralized training with local interactions can still improve reward performance and satisfy safe constraints.

2 Preliminaries

2.1 Constrained Markov game

Consider a safe MARL problem subject to multiple constraints, where each agent are associated with an underlying undirected graph G =( N , E ) . Here, N = { 1 , . . . , n } is the set of n agents and E ⊂ N × N is the set of edges. The problem can be formulated as a constrained Markov game 〈N , S , A , P, ρ 0 , γ, R , C , c 〉 . S = × i ∈N S i and A = × i ∈N A i are the state and action spaces, which are the product of local spaces; global state s = (s 1 , . . . , s n ) and joint action a = (a 1 , . . . , a n ) for any s ∈ S and a ∈ A . P : S × A × S - → R is the probabilistic transition dynamics function, which

satisfies the Dobrushin condition [13] as follows:

where z j = (s j , a j ) and z ′ j = (s ′ j , a ′ j ) represent two different state-action pairs of the agent j respectively, and z -j represents the state-action pair of the agent other than j . The value of W ij reflects the extent to which the local transition probability of agent i is affected by the state and action of agent j . ρ 0 is the initial state distribution, γ ∈ [0 , 1) is the discount factor. R : S × A -→ R is the joint reward function, C = { C i j } i ∈N 1 ≤ j ≤ m i is the sets of cost functions (every agent i has m i cost functions) of the form C i j : S i ×A i -→ R , and finally the set of corresponding cost values is given by c = { c i j } i ∈N 1 ≤ j ≤ m i .

At each timestep t , every agent i is in a state s i t , and takes an action a i t according to its policy π i = (a i | s i t ) . Together with other agents actions, it gives a joint action a t = (a 1 t , . . . , a n t ) and the joint policy π = ∏ n i =1 π i (a i | s i t ) . The agents receive the reward R ( s t , a t ) , meanwhile each agent i pays the costs C i j ( s i t , a i t ) , ∀ j = 1 , . . . , m i , and all agents have a joint goal, i.e., maximizing the expected total reward of

meanwhile satisfying every agent i 's safety constraints, written as

2.2 Spatial correlation decay

Exponential decay property [13, 14], also known as spatial correlation decay, is a powerful property associated with local interactions, which says that the impact of agents on each other decays exponentially in their graph distance. More information about spatial correlation decay is presented in Appendix B.1. Here, inspired by [15], we make the following two assumptions for the spatial correlation of the transition dynamics and policies. We use the notation π i ( ·| s N i κ ) for κ -hop policies, where s N i κ represents the state of agent i 's κ -hop neighbors. It may be replaced with π i κ for simplicity when it is clear from context.

Assumption 2.1. (Spatial Decay of Correlation for the Dynamics) Assume that there exist β > 0 in (1), for any agents i, j ∈ N , such that

where d ( i, j ) represents the distance between agent i and agent j , and ζ ∈ [0 , 2 /γ ) is a constant. Assumption 2.2. (Spatial Decay of Correlation for the Policies) Assume that there exist ξ, β ≥ 0 such that for any agent i ∈ N , s N i κ ∈ S N i κ , s N -i κ , s ′ N -i κ ∈ S N -i κ , one have

Assumption 2.2 reveals how much information is lost compared with access to the global state and allows us to consider a policy class with the necessary properties for the optimal policy under Assumption 2.1. More information is stated in Appendix B.2.

3 Scalable constrained policy optimization

This section develops a novel scalable and theoretically-justified multi-agent constrained policy optimization method and proposes a practical algorithm, i.e., Scal-MAPPO-L, by parameterizing each agent's policy. Specifically, we first quantify the maximum information loss regarding the

advantage truncation based on the spatial correlation decay property of the transition dynamics and policies. Then, the rigorous bounds of the trust region method and the bounds of the truncated advantage function are integrated to provide a new local policy optimization objective for each agent. Further, we prove that the safety constraints and the joint policy improvement can be guaranteed when updating the local police with a sequential update scheme, in which the policy update only depends on its action and the state of its κ -hop neighbors for each agent.

3.1 Truncated advantage function estimator

For a standard safe MARL, the state-action value function (the definition can be seen in Appendix C.1) and advantage function of agent i yield that

where s represents the global state, a -i represents the actions of all other agents, and Q j,i π ( s , a j , a i ) represents the state-action value function of agent i and agent j . Then, updating agents' policies with a sequential update scheme [16], the multi-agent joint advantage function A π ( s , a ) can be written as a sum of sequentially unfolding multi-agent advantages of individual agents, as stated by the following lemma.

Lemma 3.1. (Multi-agent advantage decomposition). For any action a i , i ∈ N , and the state s ∈ S , the following identity holds

Similar result to Lemma 3.1 can be seen in [10], and the proof is reported in Appendix C.2. Specifically, based on the multi-agent advantage decomposition in Lemma 3.1, the "surrogate" return is given as follows.

Definition 3.2. Let π be a joint policy, ¯ π 1: i -1 be some other joint policy of agents 1 : i -1 , and ˆ π i be a policy of agent i . Then, the surrogate return can be defined as

Building on Lemma 3.1 and Definition 3.2, one can obtain

Further, recalling Assumption 2.1 and Assumption 2.2, we can quantify the maximum information loss regarding the advantage function as stated by the following proposition.

Proposition 3.3. For any agent i ∈ N , let the parameters ( η, φ ) = ( ξγζ 1 -γζ , e -β ) . If Assumption 2.1 and Assumption 2.2 hold, for any z N i κ = ( s N i κ , a N i κ ) ∈ S N i κ ×A N i κ , the exponential decay property of the advantage function holds, i.e., we have

Proposition 3.3 shows that when the transition dynamics and policies correlation satisfy the exponential correlation decay property, the advantage functions also have exponential decay dependence on the states and actions of the more distant agents. The proof of Proposition 3.3 is reported in Appendix C.3. In addition, based on this proposition, we can obtain the following corollary.

Corollary 3.4. For any agent i ∈ N , let the parameters ( η ′ , φ ) = ( M i ξ 1 -γ + (2+ ξ ) γζ 1 -γζ , e -β ) , M i is a constant. If Proposition 3.3 holds, the exponential decay property of the surrogate return holds, i.e., we have

The proofs of Corollary 3.4 is reported in Appendix C.4.

Corollary 3.4 shows that the approximation error of L i π i κ ( ¯ π i κ ) decreases exponentially with κ when the truncated advantage functions are bounded. The main advantage of using the estimator L i π i κ ( ¯ π i κ ) lies in that every agent i only needs to know the action and state of its κ -hop neighbors, which can significantly reduce the communication burden and expand its application scenarios.

3.2 Scalable constrained policy optimization

With the Definition 3.2, we see that Lemma 3.1 allows for decomposing the joint surrogate return L π ( ¯ π ) glyph[defines] E s ∼ ρ π , a ∼ ¯ π [ A π ( s , a )] into a sum over surrogates of L 1: i π ( ¯ π 1: i -1 , ˆ π i ) . Then, combining the rigorous bounds of the trust region method [17] and the bounds of the truncated advantage function, we can obtain the following proposition.

Proposition 3.5. Let π and ¯ π be joint policies. Let each agent i ∈ N sequentially solves the following optimization problem:

where ( η ′ , φ ) = ( M i ξ 1 -γ + (2+ ξ ) γζ 1 -γζ , e -β ) , ν i κ = 2 γ max s N i κ , a i ∣ ∣ ∣ A i π i κ (s N i κ , a i ) ∣ ∣ ∣ (1 -γ ) 2 , and D max KL ( π i κ | ˆ π i κ ) = max s N i κ D KL ( π i ( · | s N i κ ) , ˆ π i ( · | s N i κ ) ) , then the resulting joint policy ¯ π will improve the expected return, i.e.,

The proof of Proposition 3.5 is reported in Appendix C.5. Similarly, by generalizing the result about the surrogate return in Equation (12), we can derive how the expected costs change when the agents update their policies. Specifically, we provide the following corollary.

Corollary 3.6. Let π and ¯ π be joint policies. For any agent i ∈ N and its cost index j ∈ { 1 , . . . , m i } , the following inequality holds

The proofs of Corollary 3.6 is reported in Appendix C.6.

From (14), we can derive that the lower bound for the difference between the new joint policy ¯ π and the old joint policy π in terms of expected return can be decomposed into a cumulative sum of local surrogate TRPO policy objectives. From (15), we can derive the upper bound for the new joint policy ¯ π , which can be used to restrict agents only to choose safe actions. Therefore, we use the objective, i.e., maximize the lower bound for the reward performance and minimize the upper bound for the safety constraints with a proper update size, as a surrogate for each agent. Then, we can obtaine the following theorem.

Theorem 3.7. The joint policy π has the monotonic improvement property, J ( ¯ π ) ≥ J ( π ) , as well as it satisfies the safety constraints, J i j ( ¯ π ) ≤ c i j , for any agent i ∈ N and its cost index j ∈ { 1 , . . . , m i } , when the policy is updated by following a sequential update scheme, that is, each agent sequentially solves the following optimization problem:

The proof of Theorem 3.7 is reported in Appendix C.7. It assures that if one follows (16) to update policies, agents will not only explore safe policies independently; meanwhile, every new policy will be guaranteed to result in performance improvement. It is worth mentioning that these two properties hold only under the condition that the only policy update restriction, i.e., ¯ π i κ ∈ ¯ Π i κ , is satisfied; this is due to the KL-penalty term in every agent's objective, i.e., ν i κ D max KL ( π i κ , ¯ π i κ ) , as well as the constraints on cost surrogates.

3.3 Algorithm

In this section, we focus on how to practically implement policy updates in Theorem 3.7 for each agent. Specifically, we parameterize each local policy π i θ i κ by a neural network with parameter θ i κ . At each policy update, every agent i maximizes its surrogate return subject to surrogate cost constraints and a form of expected KL-divergence constraint ˜ D KL ( π i κ , ¯ π i κ ) ≤ δ i κ , which avoids computing KL-divergence at every state. Then, we introduce a scalar variable λ i for any agent i ∈ N and convert the constrained optimization problem from (16) into a min-max optimization problem with Lagrangian multipliers by subsuming the cost constraints. As such, the new optimization problem for any agent i ∈ N is as follows:

where λ i 1: m i is a scalar variable, θ i κ is a parameter of neural network, and d i u is the cost-constraining value for agent i .

Further, denoting

then the optimization problem in (17) can be rewritten as

To alleviate the complications caused by computing the KL-divergence constraint, we simplify it by adopting the PPO-clip objective [18], i.e., replacing the KL-divergence constraint with the clip operator and updating the policy parameter with first-order methods. The final optimization problem takes the form

where the clip operator replaces the policy ratio with 1 + glyph[epsilon1] , or 1 -glyph[epsilon1] , depending on whether its value is below or above the threshold interval. As such, agent i can learn within its trust region by updating

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To summarize, we give a procedure for each agent i , name Scalable MAPPO-Lagrangian (ScalMAPPO-L), and provide its pseudocode (Algorithm 1) in Appendix C.8. The algorithm has a simple idea that each agent independently optimizes the surrogate objective (20), which only depends on its action and the state of its κ -hop neighbors for each agent. In the actual execution, some approximations of the surrogate objective are employed, the same as the MAPPO-L [10]. Most of these approximations are traditional practices in RL, yet they may make it impossible for the practical algorithm to rigorously maintain the theoretical guarantees in Theorem 3.7.

4 Experiments

In this section, we evaluate our method via several numerical experiments. Our experiments aim to answer the following questions: First, how does the cost and reward performance of Scal-MAPPO-L compare with existing methods on challenging multi-agent safe tasks? Second, how does the different κ affect the performance of Scal-MAPPO-L, and could the advantage truncation effectively alleviate computational load?

4.1 Experimental setup

Safe MAMuJoCo [10] is an extension of MAMuJoCo [19], which preserves the agents, physics simulator, background environment, and reward function and comes with obstacles, like walls or pitfalls. To answer the first question, we compare our method against the other PPO family algorithms, i.e., IPPO [20], HAPPO [16], and MAPPO-L [10] and choose three games from Safe MAMuJoCo: Safe ManyAgent Ant task with 2 agents (2 × 3), 3 agents (3 × 2) and 6 agents (6 × 1) to evaluate their performance. Concerning the second question, we choose three games with different tasks and agent numbers from Safe MAMuJoCo: Safe ManyAgent Ant task with 6 agents ( 6 × 1 ), Safe Ant task with 8 agents ( 8 × 1 ), and Safe Coupled HalfCheetah task with 12 agents ( 12 × 1 ). We train Scal-MAPPO-L with the same network architecture and hyperparameters as the original MAPPO-L implementation. All reported results are averaged over three or more random seeds, and the curves are smooth over time.

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4.2 Results

Comparisons with baselines: Figure 1 shows the cost and reward performance of Scal-MAPPO-L and other PPO family algorithms on three Safe ManyAgent Ant tasks, where each agent in ScalMAPPO-L is set to access the state of about half of the agents by adjusting the value of κ . Specifically, κ = 1 in Safe ManyAgent Ant ( 2 × 3 ), κ = 2 in Safe ManyAgent Ant ( 3 × 2 ), and κ = 3 in Safe ManyAgent Ant ( 6 × 1 ). From Figure 1, we can see that compared to IPPO and HAPPO, on all three tasks, both Scal-MAPPO-L and MAPPO-L have fewer constraint violations and good performance (in terms of reward), i.e., they keep their explorations within the feasible policy space and quickly learn to satisfy safety constraints, which show that the safe learning algorithm is effective. Moreover, it should be further pointed out that Scal-MAPPO-L only accesses half of the state information on all tasks; it exhibits almost identical performance and constraint violations with MAPPO-L (which accesses the global state). This means that the sensitivity of each agent to the states and actions perturbations of distant agents is minimal, and Scal-MAPPO-L is effective. More experimental results are in Appendix D.

Performance with different κ : Figure 2 shows the performance of Scal-MAPPO-L in different environments with varying values of κ , where MAPPO-L accesses the global state. We have noticed that the algorithm's performance is consistently the lowest, and the cost is nearly the highest when κ = 1 . However, when the truncation with κ > = 3 , i.e., each agent has access to the states of at least two neighbors, we can observe that the performance of Scal-MAPPO-L improves considerably and can approach or even outperform MAPPO-L in some environments, such as κ = 6 in the Safe Ant task ( 8 × 1 ). This may be due to the fact that the impact of far-away agents' states and actions on the agent's decision is almost negligible in many cases. However, for algorithms with global communication, such as MAPPO-L, the difficulty of extracting useful information from many messages may lead to lower performance. Overall, these results underscore the efficiency of Scal-MAPPO-L since it employs a smaller communication radius that can significantly reduce the computation.

5 Related work

5.1 Safe RL

Safety is one of the bottlenecks preventing RL use in real-life applications, such as physical robotics [21], medical applications [22] and autonomous driving [23]. It has become a research hotspot in recent years and a growing number of safe RL approaches, such as primal-dual methods [24], formal methods [25], Lyapunov methods [26], Gaussian processes methods [27], and safety-augmented

methods [28], have been developed. However, when it comes to multi-agent systems, a great challenge is exacerbated by policy conflicts caused by multiple agents interacting within a shared environment and learning simultaneously. In other words, each agent has to not only satisfy its safety constraints but also consider the conflicts between its safety constraints and maximization reward as well as the safety constraints of others so that their joint behaviors have a safety guarantee. In order to address the above issue, CMIX [9] and MAPPO-L [10] have been proposed with the in-depth study of MARL. These algorithms follow the centralized training and decentralized execution (CTDE) framework [29, 30, 16], which learns the centralized value function by introducing the global state. Unfortunately, the global coupling arising from agents' safety constraints and the exponential growth of the state-action space size make the usability in communication or computing resource-constrained systems and the scalability of these algorithms in larger multi-agent systems become a bottleneck, limiting their applicability. Recent works [11, 12] have provided some theoretical results to avoid these shortcomings. However, most of these methods fail to ensure both safety guarantee and joint policy improvement under a decentralized learning framework under a decentralized learning framework, which motivates us to investigate a new scalable and theoretically-justified safe MARL method.

5.2 Centralized training

In cooperative MARL settings, the training of agents can be broadly divided into two paradigms, namely centralized and decentralized [31]. The centralized training paradigm describes agent policies updated based on mutual information, which can be further differentiated into the centralized and decentralized execution framework. Centralized training and centralized execution (CTCE) utilize the centralized evaluator and executor to learn the joint policy of all agents [32, 18]. The obvious flaw is that its applicability is limited because its implementation requires the premise that instantaneous and unconstrained information exchange between agents. Recently, centralized training and decentralized execution (CTDE) has become the most popular framework [30, 20, 16, 10], since the fact that it addresses the non-stationarity issue with the centralized value function, and removes the dependency on global state and actions during execution. Many experiment results demonstrate state-of-the-art performance on challenging tasks, such as unit micromanagement in StarCraft II [33]. However, although this framework does not require agents to access the global state during execution, the reliance on the global state only during training still poses a significant barrier to real-world applications, especially in scenarios where communication and computational resources are constrained [34, 35].

5.3 Decentralized training

In a decentralized learning paradigm, each agent learns independently and accessses local observations rather than the global state; the idea is direct, comprehensible, and easy to realize in practice [36, 34]. There are two mainline research approaches concerning decentralized learning in the existing literature. One line of research pursues fully decentralized learning, such as independent Q-learning (IQL) [37, 38] and independent actor-critic (IAC) [39, 20], which make agents directly execute the single-agent Q-learning or actor-critic algorithm individually. Another line of research allows agents to establish rational local communication networks, such as setting certain distance or neighbor graphs [40, 41], which is also known as networked MARL. Communication networks expand agents' perceptual capabilities and mitigate, to some extent, the decision conflicts or errors caused by partial observability. However, it is worth noting that each agent's decision violates the stationary condition of the Markov Decision Process (MDP) in both lines of research, even though they achieve good experimental results on a collection of benchmark tasks. It poses a significant challenge to the convergence analysis of algorithms in the short term. Recently, motivated by good experiment performance, some studies have tried to provide theoretical support for these phenomena. To mention a few, Qu et al. [42] introduced the spatial correlation decay property into the field of MARL and carried out a series of fundamental results [15, 43, 12], which broadened the research avenues of scalable MARL. However, all of these studies mainly focus on (natural) policy gradient methods with average rewards or general utilities and have not yet been combined with trust region methods, which rigorously enable RL agents to learn monotonically improving policies. Furthermore, only recent research [12] considers both safety and scalability for MARL. Our results build upon the scalable MARL family of works [42, 15, 43, 12] and PPO-based (TRPO-based) MARL family of works [16, 10].

6 Conclusion

Safety is a tremendous challenge for MARL when applied to real-world scenarios. In this paper, we quantize the approximation errors arising from policy implementation and advantage truncation and then derive a novel lower bound for joint policy improvement and an upper bound for the safety constraints for every agent. Furthermore, we propose a novel scalable and theoretically justified multi-agent constrained policy optimization method that follows a sequential update scheme to optimize κ -hop policies. Finally, we introduce a practical constrained policy optimization algorithm called Scal-MAPPO-L and experimentally validate the effectiveness of the proposed algorithm on a collection of benchmark tasks.

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A Preliminary lemmas

Before proving propositions, corollaries, and theorems, we need a series of intermediate results as a foundation. Results similar to Lemmas A.1 and A.2 can be found in Chapter 8 of [44], Lemma A.3 is an extension of results from [15], Lemma A.4 is an extension of Lemma A.2 found in [12]. We state these lemmas and provide the corresponding proofs for completeness as follows.

Lemma A.1. Let f : S → [ m,M ] , where S = × i ∈N S i and m,M ∈ R . For every i ∈ N , let µ i and ν i be two distributions on S i . Let µ and ν be the respective product distributions. Let δ i ( f ( s )) = sup s i , s -i , s ′ i ∣ ∣ f ( s i , s -i ) -f ( s ′ i , s -i )∣ ∣ . Then, one have

Proof. We prove Lemma A.1 by induction. Note that

is an equivalent formulation of the total variation distance [45].

For |N| = 1

As induction assumption, assume that Lemma A.1 holds for |N| > 1 . Then, one have

where ˜ f ( s ∼ ν

By induction assumption, one have

Since glyph[negationslash]

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one have

which concludes the induction.

Lemma A.2. Consider a Markov Chain with state s ∈ S , where S = × i ∈N S i , and N = { 1 , . . . , n } is the set of agents. Suppose its transition probability factorizes as

Let W ∈ R n × n be a matrix whose elements respect the condition

If ∑ j ∈J e βd ( j,i ) W ij ≤ ζ , J ⊆ N , then one have sup s j , s -j , s ′ j D TV ( P i ( · | s J , s -J ) , P i ( · | s ′ J , s -J )) ≤ ∑ j ∈J W ij , (22)

and

where d ( J , i ) = min j ∈J d ( j, i )

Proof. We prove the first claim of Lemma A.2. The first claim clearly holds if |J | = 1 . As induction assumption, assume that the first claim holds for a set J . Then, it holds for J ′ = J + { k }

The second claim follows immediately, since and

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Lemma A.3. Consider the setting of Lemma A.2. For a generic value of κ , denote by ρ t and ˜ ρ t the distribution of s t with starting state, respectively, s = ( s N i κ , s N -i κ ) and ˜ s = ( s N i κ , ˜ s N -i κ ) . Then, if ∑ j ∈N e βd ( j,i ) W ij ≤ ζ , we have that D TV ( ρ i t , ˜ ρ i t ) ≤ ζ t e -βκ , ∀ i ∈ N .

Proof. We prove Lemma A.3 by induction. The case where t = 1 follows from Lemma A.2. As induction assumption, assume that Lemma A.3 holds for t . Then, one have

where we used Lemma A.1 in the first inequality.

Lemma A.4. Consider the setting of Lemma A.2. Let P ( s | s ) = P ( s = s | s = s ) and

Proof. We prove Lemma A.4 by induction. The claim holds for t = 1 ,

As induction assumption, we assume that the claim holds for t . Then, using Lemma A.1,

and using the inverse triangle inequality,

which concludes the induction.

B Supplementary materials for Section 2

B.1 Spatial correlation decay

Exponential decay property [13, 14], also known as spatial correlation decay, is a powerful property associated with local interactions, which says that the impact of agents on each other decays exponentially in their graph distance. Over the past decades, many researchers have utilized spatial correlation property to design scalable, distributed algorithms for optimization and control problems in scenarios such as epidemics [46] and wireless communication [47]. Inspired by the studies mentioned above, a recent line of work [48] has formally considered spatial decay of correlation assumptions and proposes a method that finds nearly optimal local policies. An application [49] with the same principles adopts the setting of mean-field MARL [50], which proposes an actor-critic algorithm with global convergence. However, unlike the mean-field setting, which requires an agent's transition scheme to be only affected by the mean effect from its neighbors and effective only when agents are homogeneous, we allow each agent to have different transition probabilities and local policies.

B.2 Regarding Assumptions 2.1 - 2.2

Assumption 2.1 portrays a common phenomenon: the transition dynamic of each agent is exponentially less sensitive to perturbations of the states and actions of more distant agents. This is commonly seen in scenarios involving wireless communication, epidemics, traffic, and so on [46, 47]. Assumption 2.2 imposes a design constraint for the policy class that encodes a weaker correlation decay property than the assumptions on the nature of Assumption 2.1. Moreover, Assumption 2.2 reveals how much information is lost compared with access to the global state and allows us to consider a policy class with the necessary properties for the optimal policy under Assumption 2.1. Below, we use a mathematical example to illustrate the relationship between the two assumptions.

Mathematical example: Firstly, we start from Assumption 2.1, letting ˜ κ = max i,j ∈N d ( i, j ) be the maximum distance between agent i and agent j . Define a set of differentiable functions { f κ : S N i κ ×A i →K| 0 ≤ κ ≤ ˜ κ } , where K ⊂ [ -K,K ] , K > 0 , and a set of parameters { α κ ≥ 0 | 0 ≤ κ ≤ ˜ κ } . Then, for each agent i , one have

By tuning the parameters α κ , we can make any policy belonging to this policy class respect Assumptions 2.2, as we show in the following. Let κ ∈ { 0 , . . . , ˜ κ } , s , ˜ s ∈ S be such that s N i κ = ˜ s N i κ , then one have

Denote that ( ξ, β ) = ( 2 Ke 2 K ˜ κ ∑ ˜ κ κ ′ = κ +1 α κ ′ , 2 K ) , and setting the parameters { α κ ′ } κ ′ ∈{ κ +1 ,..., ˜ κ } small enough ensures that the policy respects Assumption 2.2.

Remark B.1 . The mathematical example illustrates the relationship between the Assumptions 2.1 and 2.2. It is evident from this mathematical example that Assumption 2.2 necessarily holds when Assumption 2.1 holds and the parameters ξ and β satisfy certain conditions. However, for the sake of more concise presentation, we treat it as a separate assumption.

Remark B.2 . When Assumption 2.1 holds, the numerical example can provide a reference basis for selecting the values of the parameters in Assumption 2.2. However, accurately determining the spatial decay of correlation for the dynamics remains a challenging engineering task. In this paper, we empirically adopt conservative values.

Remark B.3 . Assumption 2.2 implies that multi-agent environments must satisfy the requirement that the impact from far-away agents' states and actions is almost negligible for the agent's decision; in other words, an action of an agent has an instantaneous effect on the system only locally. We believe that this formulation realistically describes most multi-agent interactions in the real-world. Take multi-vehicle transportation as an example. For a vehicle traveling on the road, a far-away vehicle taking different actions or being in a different state will affect itself shortly thereafter, but the impact on the current policy is minimal. More examples are seen in wireless communication, epidemics, traffic, and other scenarios [46, 47].

Remark B.4 . It is worth noting that the assumption of spatial correlation decay is not in direct conflict with well-known phenomena, e.g., Butterfly Effect, since two seemingly unrelated things can also have a significant impact on each other, generally occurring in different time domains.

C Supplementary materials for Section 3

C.1 Basic definitions

Regarding the state value function and the state-action value function, we give the following definitions.

Definition C.1 . We define the state value function and the state-action value function in terms of reward as

Based on C.1, one can expand to derive

Definition C.2 . We define the j th state cost value function and state-action cost value function for agent i as follows

C.2 The proof of Lamma 3.1

Proof. We write the multi-agent advantage function as in its definition, and then expand it in a telescoping sum.

C.3 The proof of Proposition 3.3

Proof. Let s , ˜ s ∈ S , a , ˜ a ∈ A , such that for any agent i ∈ N , s N i κ = ˜ s N i κ and a N i κ = ˜ a N i κ . According to Equation (7), when only the state and action of the far-away agent are different, one have

Next, we analyze ∣ ∣ Q i π ( s , a ) -Q i π ( ˜ s , ˜ a ) ∣ ∣ and ∣ ∣ V i π ( s ) -V i π ( ˜ s ) ∣ ∣ separately.

Firstly, for ∣ i i ∣

where ρ i t and ˜ ρ i t are the distributions at time t with starting point ( s , a ) and ( ˜ s , ˜ a ) , respectively. We use the result in Lemma A.3 to bound D TV ( ρ i t , ˜ ρ i t ) . The structure of our MDP implies that:

Then, if Assumption 2.1 holds, the requirements of Lemma A.3 are satisfied, one have D TV ( ρ i t , ˜ ρ i t ) ≤ ζ t e -βκ and

where ζ is defined in Assumption 2.1.

Let

and the MDP satisfies the condition of Lemma A.4, one can obtain

Secondly, building on Assumption 2.2 and Lemma A.4, we analyze ∣ ∣ V i π ( s ) -V i π ( ˜ s ) ∣ ∣ can further obtain

Then, bringing (32) and (33) into (31), we have

Finally, denoting ( η, φ ) = ( (2+ ξ ) γζ 1 -γζ , e -β ) , we can obtain the Proposition 3.3.

C.4 The proof of Corollary 3.4

Proof. Firstly, according to Lamma 3.1 and Definition 3.2, the following result holds when each agent adopts a sequential update scheme to optimize policy, i.e., we have

Then, based on Assumptions 2.2, we have

According to (34), we have

Then, bringing (36) and (37) into (35), we have

Finally, denoting ( η ′ , φ ) = ( M i ξ 1 -γ + (2+ ξ ) γζ 1 -γζ , e -β ) , we can obtain the Corollary 3.4.

C.5 The proof of Proposition 3.5

Proof. From (12), we can obtain -η ′ φ κ ≤ L 1: i π ( ¯ π 1: i -1 , ¯ π i ) -L i π i κ ( ¯ π i κ ) ≤ η ′ φ κ . By the trust region theorem in Theorem 1 from [17], we have

Then, when each agent sequentially solves the following optimization problem:

where ( η ′ , φ ) = ( M i ξ 1 -γ + (2+ ξ ) γζ 1 -γζ , e -β ) , ν i κ = 2 γ max s N i κ , a i ∣ ∣ ∣ A i π i κ (s N i κ , a i ) ∣ ∣ ∣ (1 -γ ) 2 , and D max KL ( π i κ | ˆ π i κ ) = max s N i κ D KL ( π i ( · | s N i κ ) , ˆ π i ( · | s N i κ ) ) , we have J ( ¯ π ) -J ( π ) ≥ ∑ n i =1 ( L i π i κ ( ˆ π i κ ) -η ′ φ κ -ν i κ D max KL ( π i κ | ˆ π i κ ) ) .

C.6 The proof of Corollary 3.6

Proof. Firstly, by generalizing the result about the return in (14), one can derive how the expected costs change when the agents update their policies. Inspired by [10], we provide the following lemma.

Lemma C.3 . Let π and ¯ π be joint policies. Let i ∈ N be an agent, and j ∈ { 1 , . . . , m i } be an index of one of its costs. The following inequality holds

Proof. From the upper bound version of Theorem 1 of [17] applied to joint policies ¯ π and π , we conclude that

where α = D max TV ( π 1: i , ¯ π 1: i ) = max s D TV ( π 1: i ( · | s ) , ¯ π 1: i ( · | s )) .

Then, using Pinsker's inequality D TV ( p, q ) 2 ≤ D KL ( p, q ) / 2 , we obtain

where D max KL ( π 1: i , ¯ π 1: i ) = max s D KL ( π 1: i ( · | s ) , ¯ π 1: i ( · | s )) .

Notice that we have E s ∼ ρ π , a 1: i ∼ ¯ π 1: i [ A i j, π ( s , a i )] = E s ∼ ρ π , a i ∼ ¯ π i [ A i j, π ( s , a i )] as the action of agents other that i do not change the value of the variable inside of the expectation. Furthermore,

Setting ν i j = 2 γ max s , a i | A i j,π ( s , a i ) | (1 -γ ) 2 , we finally obtain

Secondly, from (12), we can obtain -η ′ φ κ ≤ L 1: i π ( ¯ π 1: i -1 , ¯ π i ) -L i π i κ ( ¯ π i κ ) ≤ η ′ φ κ . By generalizing the result, we can obtain -η ′′ φ κ ≤ L 1: i j, π ( ¯ π 1: i -1 , ¯ π i ) -L i j,π i κ ( ¯ π i κ ) ≤ η ′′ φ κ . Further, we can derive the upper bounds about surrogate cost

C.7 The proof of Theorem 3.7

Proof. Based on the conclusions in Proposition 3.5 and Corollary 3.6, we can derive that in order to realize reward performance improvement and satisfy safety constraints, agents have to sequentially maximize their surrogate returns and ensure that their surrogate costs stay below the corresponding safety thresholds. Meanwhile, they have to constrain the policy search to small local neighborhoods (w.rt, max-KL distance). Therefore, the size of KL constraint in Equation (16) should be set as

where h ∈ N i κ is the κ -hop neighbors of agent i , and j ∈ { 1 , . . . , m h } is its cost index.

Note that δ 1 κ is guaranteed to be non-negative if π satisfies safety constraints; that is because then c h j ≥ J h j ( π ) for all h ∈ N , and j ∈ { 1 , . . . , m i } , and the set { h | h < i } is empty.

This formula for δ i κ , combined with Lemma 3.1, assures that the policies π i κ within δ i κ max-KL distance from π i κ will not violate other agents' safety constraints, as long as the base joint policy π did not violate them (which assures δ 1 κ ≥ 0 ). To see this, for every h = 1 , . . . , i -1 , and j = 1 , . . . , m h , we have

which implies

By Corollary 3.6, the left-hand side of the inequality (39) is an upper bound of J h j ( ¯ π 1: i -1 , π i ) , which implies that the update of agent i does not violate the constraint of J h j . The fact that the constraints of J h j for h ≥ i +1 are not violated, i.e.,

c h j -J h j ( π h κ ) -ν h j,κ ∑ i -1 l =1 D max KL ( π l κ , ˆ π l κ ) , when the policy is updated by following a sequential update scheme, that is, each agent sequentially solves the following optimization problem:

the joint policy π has the monotonic improvement property, J ( ¯ π ) ≥ J ( π ) , as well as it satisfies the safety constraints, J i j ( ¯ π ) ≤ c i j , for any agent i ∈ N and its cost index j ∈ { 1 , . . . , m i } .

C.8 Algorithm

In this subsection, we provide the main pseudocode for Scalable MAPPO-Lagrangian (Scal-MAPPOL), as outlined in Algorithm 1.

Algorithm 1 Scalable MAPPO-Lagrangian

Input: Stepsizes α θ , α λ , batch size B , number of agents n , episodes Z , steps per episode T , discount factor γ , parameter κ .

Initialize: Actor networks θ 1 κ, 0 , . . . , θ n κ, 0 , V-value network χ 1 κ, 0 , . . . , χ n κ, 0 , V-cost networks { φ i j, 0 } i ∈N 1 ≤ j ≤ m i , ∀ i ∈ N , j ∈ 1 , . . . , m i , Replay buffer B .

1: for z = 0 , 1 , . . . , Z -1 do

2: Collect a set of trajectories by running the policies π θ 1 κ , . . . , π θ n κ .

3: Push transitions {( o i t , a i t , o i t +1 , r i t ) , ∀ i ∈ N , t ∈ T } into B .

4: Sample a random minibatch of B transitions from B .

5: for i = 1 : n do

6: Initialize a policy parameter θ i κ, 0 and Lagrangian multipliers λ i j , ∀ i ∈ N , j ∈ 1 , . . . , m i .

7: Compute advantage function ˆ A i ( s , a i ) and cost advantage functions ˆ A i j ( s , a i ) .

8: Compute the parameters η ′ , η ′′ , φ, ν i κ and ν i j,κ , ∀ j ∈ { 1 , . . . , m i } .

9: Compute the radius of the KL-constraint δ i κ .

10: Compute the advantage function in (18).

11: Update policy according to (20).

12: Update V-value network and V-cost networks.

13: end for

14: end for

The algorithm has a simple idea that each agent independently optimizes the surrogate objective, which only depends on its action and the state of its κ -hop neighbors for each agent. In the actual execution, we adopt the surrogate objective (20) instead of (16). It actually uses some approximations for the decentralized surrogate objective, the same as the MAPPO-L [10]. Most of these approximations are traditional practices in RL, yet they may make it impossible for the practical algorithm to rigorously maintain the theoretical guarantees in Theorem 3.7. However, we need to argue that we should go one step further and provide a decentralized surrogate for decentralized learning with a convergence guarantee. We believe and expect that a better practical method can be found based on this objective in future work.

D Supplementary materials for Section 4

D.1 Additional experimental results

In this paper, we compare the algorithm of our proposed (i.e., Scal-MAPPO-L in Algorithm 1) against other PPO family algorithms on several safe MARL tasks to evaluate their performance. Here, we provide some additional experimental results, which are illustrated in Figures 3-4.

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Remark D.1 . It is worth pointing out that, in our code, unlike the original, the global state consists of a patchwork of each agent's ID and the κ -hop information rather than a long state vector. This is the main reason of the difference in performance from the original paper. As we consider decentralized learning, the agents in the experiments do not use parameter-sharing. In all experiments, the network architectures and common hyperparameters of our algorithm and MAPPO-L are the same for a fair

comparison. All reported results are averaged over three or more random seeds, and the curves are smooth over time.

D.2 The computer resources and computational complexity

Computation resources : We executed our code on a computer with NVIDIA GeForce RTX 4090 (GPU) and Intel Core i9-13900K (CPU).

Computational complexity : The computational complexity of Scal-MAPPO-L (ours) is O(TNMHP), where T denotes the number of steps, N denotes the number of agents, M denotes the number of constraints, H denotes the number of PPO-Epoch, and P denotes the number of policy parameters.

Besides, we test the running time of Scal-MAPPO-L on Safe Manyagent Ant 6 × 1 , Safe Ant 8 × 1 , and Safe Coupled HalfCheetah 12 × 1 . The running steps are 1 × 10 7 in each environment. When the parameter κ is maximized, the algorithm's average wall-clock times are 8 . 43 h , 9 . 28 h , and 11 . 65 h , respectively. It is worth noting that the wall-clock times do not significantly down when κ gradually decreases. This may be due to the fact that we have yet to consider the process of sending and receiving information realistically. However, based on the successful research conducted in the field of communication [51, 52], it is evident that algorithms requiring less communication undoubtedly have an advantage in terms of reducing communication burden and enhancing applicability.

E The discussion of limitations and impacts

E.1 Limitations

This paper is centered on theoretical analysis and also contains practical algorithms and simulation verification. The main results in the paper characterize the proposed method's performance in terms of safety constraints and joint policy improvement. Below, we discuss the limitations of the proposed approach for both theory and experiment aspects as follows:

1) Our theoretical results are based on the two assumptions about the spatial decay of correlation for the dynamics and the policies in Assumption 2.1 and Assumption 2.2. Our conclusions may be useless when such assumptions do not hold, e.g., the decisions of each agent are non-negligibly related to the decisions of all other agents. However, fortunately, existing works [46, 47] have shown that many real-world situations satisfy both assumptions, so our study is still important and meaningful.

2) Our experiments show that Scal-MAPPO-L, with communication between a small number of neighbors, outperforms MAPPO-L in some cases, which we would like to see because it implies fewer communication requirements. However, we have yet to develop an equilibrium relationship between the amount of communication and the performance, which we will focus on next.

E.2 Broader impacts

This paper presents work that aims to advance the field of RL, especially safe MARL. Our work has many positive societal impacts, such as providing a theoretical foundation for scalable Safe MARL, none of which we feel must be specifically highlighted. There are no negative societal impacts on our work.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the problem of constrained MARL in a cooperative setting and focus on the decentralized learning settings without global observability. The paper proposes a constrained policy optimization method and its practical version, Scal-MAPPO-L. Theoretical results are established for the dynamics/policy truncation and the trust-region subproblems. The effect of the proposed method is validated through numerical experiments.\n\nSTRENGTHS:\nThe paper is well-organized, and complete in structures. \n\n* In terms of theoretical contribution, the paper derives a monotone improvement property in the exact setting, i.e., when no parameterization is involved. The empirical algorithm, i.e., Scal-MAPPO-L, is proposed.\n\n* In terms of numerical experiments, the paper performs reasonable experiments to compare Scal-MAPPO-L with other PPO family algorithms (with code provided). The experimental results demonstrate Scal-MAPPO-L exhibits decent performance.\n\nWEAKNESSES:\nMy major concern about the paper is regarding the novelty of the paper given the existing literature [10]. The paper shares many similarities with [10] (Safe multi-agent reinforcement learning for multi-robot control), including presentation and theoretical results. Although I understand that [10] considers a centralized setting with full observation, and the setting considered here is decentralized, I am not sure about the technical contribution of the paper beyond leveraging the spatial decay of correlation assumption.\n\nAlso, the introduction of the spatial decay of correlation is somewhat abrupt. The author seems did not mention anything about the graph structure for the agents before introducing the assumption.\n\nQUESTIONS:\nI mainly have the following questions.\n\n* Could the author make further clarifications about the technical contribution of the paper against [10]. What are the main difficulties met when extending the results in [10] to the current setting (after imposing the spatial decay of correlation assumption).\n\n* Could the author provide more justifications about why the methods proposed in the paper is better than the existing literature [11] and [12]. In the paper, the author mentions [11] imposes extra parameter-sharing constraint, which results in suboptimality. This sounds not convincing enough to me, because the method developed in this paper is also \"not optimal\". For the discussion about [12], I failed to understand the idea conveyed by the author, and hope the author could elaborate further.\n\n* The empirical algorithm Scal-MAPPO-L does not have any theoretical guarantee. I wonder is it possible to say anything about its performance guarantee (maybe under additional assumptions)?\n\nLIMITATIONS:\nThe authors have discussed the limitations in appendix, which sounds reasonable to me.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nthe work proposes a scalable version of MAPPO-L for constrained policy optimization, taking into account that decays in the inter-dependence between agents in a Markov Game projects into bounded errors while limiting the information sharing between agents. The theoretical results transfer nicely from the two starting frameworks, and the authors propose some empirical corroboration.\n\nSTRENGTHS:\n**originality**:\nThe overall combination of known results from scalable algorithms and MAPPO-L might not be that original. Having said that, it is not trivial as well, so while not surprising, I would say it is not negligible, assuming that the authors fairly address the similarities in the related works (see weaknesses).\n\n**quality**: \nthe work reaches high quality in the analysis and the empirical corroboration is mostly convincing, out of some doubts to be clarified (see Questions)\n\n**clarity**:\nthe work is overall well-written, and the concepts are generally well-described and help to understand the contributions.\n\n**significance**:\nthe work addresses an important problem, that is how to build a more scalable version of algorithms from SOTA MARL algorithms.\n\nWEAKNESSES:\n**Major**:\n\n- In the related works section, I would expect an extensive and fair discussion of the difference between this work and works on scalable MARL and MAPPO-L. The work has value in being a combination of the two, but this needs to be clarified. For example, more comments on the differences/novelties in the proofs and results from works cited as [10], [15], [18], [47].\n\n- The experimental corroboration does not provide any useful info about the difference in computational/communication burden introduced by Scalable-MAPPO-L compared to MAPPO-L. Additionally, from Figure 2, the fact that using k=1 (almost IPPO-L then) is way enough to solve the problem would rather suggest that the tasks themselves are easy. \n\n**Minor**:\n- Section 2.2 : I would strongly suggest introducing the notion of agent graph somewhere, otherwise the $S_{N_i^k}$ and the notion of distance between agents might be hard to digest from scratch.\n\n- Eq 4/5 : I would suggest distinguishing the constants $\\beta_1$ and $\\beta_2$ (in case they are not different).\n\n- Eq. 10: I believe $A^j_\\pi$ should be $A^h_\\pi$.\n\n- Eq. 13: I would specify over which class of policies are arg-maximizing.\n\n- Theorem 3.7: I would suggest refactoring the constants, they are hard to read.\n\nQUESTIONS:\n**Major**:\n- Can the authors discuss the validity of the assumptions? When would it be the case that $\\zeta \\in [0, 2/\\gamma]$? How would the condition of the policies be enforced while learning? Does it mean that the policies learned in the experiments are of the form in lines 522 in the Appendix? Are the assumptions valid in the instances of MAMUJOCO taken into account? \n\n- What do lines 259/260 stand for? what information in the environment do the authors refer to?\n\n- Code: would the authors provide a repository for the code? otherwise, the claims about the reproducibility of the work in the checklist are not very satisfied\n\n- Figure 1: with which value of k were the experiments done? Line 248 is not fully clear to me. Would the authors explain further? The performances of MAPPO-L do not match the performances of the original paper, why that?\n\n- Figure 2: Why is there a difference in performances between the two Manyagents 6x1 experiments in Figures 1 and 2?  The performances of MAPPO-L do not match the performances of the original paper, why that?\n\n- What is the meaning of the claims in lines 327, 328 about the differences with [15], [47], [12]?\n\n**Minor**:\n\n- What does $d^i_u$ in equation (17) stand for?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposed a scalable multi-agent constrained policy optimization for safe reinforcement learning. It is an extension of two previous work on safe reinforcement learning and scalable multi-agent reinforcement learning. The trust region policy updates and truncated policy/advantage function are combined to give theoritical performance bound. A practical algorithm based on PPO is also shown and the empirical results verifies the claimed performance.\n\nSTRENGTHS:\n1. The presentation is clear and easy to understand, the authors presented clearly the relation between this paper and the previous work.\n2. It is a very solid combination of the ideas in previous works, and extends the scalable multi-agent reinforcement learning idea to multi-agent safe reinforcement learning. It will be beneficial to the safe RL community.\n3. The theoretical results look correct to me.\n\nWEAKNESSES:\nThere are all minor weaknesses in this paper.\n1. The author should discuss more clearly on how the assumptions are related to previous works, for example, the Dobrushin conditions in (1) and Assumption 2.1. They both appear in the previous scalable multi-agent reinforcement learning paper [1] but in slightly different formulations.\n\n2. The experiments are only on a smaller number of agents (for 12 agents, we don't really need a scalable algorithm to handle it).\n\n[1] Guannan Qu, Yiheng Lin, Adam Wierman, and Na Li. Scalable multi-agent reinforcement 374 learning for networked systems with average reward. In NeurIPS, 2020.\n\nQUESTIONS:\nThe theoritical results look okay to me, although I didn't check all the proofs. for the experimental results,\n1. For figure 1 and 2, what is the constraint on the average episodic cost? It will be more clear to draw a horizontal line showing that.\n2. The performance is still increasing at 1e7 steps, the algorithms might not converge. Can you explain a bit?\n\nLIMITATIONS:\nThe authors addressed the limitation well.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}