# facmac_factored_multiagent_centralised_policy_gradients__d596671a.pdf

FACMAC: Factored Multi-Agent Centralised Policy Gradients

Bei Peng University of Liverpool Tabish Rashid University of Oxford Christian A. Schroeder de Witt University of Oxford

Pierre-Alexandre Kamienny Facebook AI Research Philip H. S. Torr University of Oxford Wendelin Böhmer Delft University of Technology

Shimon Whiteson University of Oxford

We propose FACtored Multi-Agent Centralised policy gradients (FACMAC), a new method for cooperative multi-agent reinforcement learning in both discrete and continuous action spaces. Like MADDPG, a popular multi-agent actor-critic method, our approach uses deep deterministic policy gradients to learn policies. However, FACMAC learns a centralised but factored critic, which combines per-agent utilities into the joint action-value function via a non-linear monotonic function, as in QMIX, a popular multi-agent Q-learning algorithm. However, unlike QMIX, there are no inherent constraints on factoring the critic. We thus also employ a nonmonotonic factorisation and empirically demonstrate that its increased representational capacity allows it to solve some tasks that cannot be solved with monolithic, or monotonically factored critics. In addition, FACMAC uses a centralised policy gradient estimator that optimises over the entire joint action space, rather than optimising over each agent s action space separately as in MADDPG. This allows for more coordinated policy changes and fully reaps the benefits of a centralised critic. We evaluate FACMAC on variants of the multi-agent particle environments, a novel multi-agent Mu Jo Co benchmark, and a challenging set of Star Craft II micromanagement tasks. Empirical results demonstrate FACMAC s superior performance over MADDPG and other baselines on all three domains.

1 Introduction

Significant progress has been made in cooperative multi-agent reinforcement learning (MARL) under the paradigm of centralised training with decentralised execution (CTDE) [27, 18] in recent years, both in value-based [41, 31, 37, 32, 44, 30] and actor-critic [23, 9, 36, 13, 8] approaches. Most popular multi-agent actor-critic methods such as COMA [9] and MADDPG [23] learn a centralised critic with decentralised actors. The critic is centralised to make use of all available information (i.e., it can condition on the global state and the joint action) to estimate the joint action-value function Qtot, unlike a decentralised critic that estimates the local action-value function Qa based only on individual observations and actions for each agent a.3 Even though the joint action-value function

Equal contribution. Correspondence to: Bei Peng <bei.peng@liverpool.ac.uk> Work done while the authors were at the University of Oxford. 3COMA learns a single centralised critic for all cooperative agents due to parameter sharing. For each agent the critic has different inputs and can thus output different values for the same state and joint action.

35th Conference on Neural Information Processing Systems (Neur IPS 2021).

these actor-critic methods can represent is not restricted, in practice they significantly underperform value-based methods like QMIX [31] on the challenging Star Craft Multi-Agent Challenge (SMAC) [35] benchmark [33, 32].

In this paper, we propose a novel approach called FACtored Multi-Agent Centralised policy gradients (FACMAC), which works for both discrete and continuous cooperative multi-agent tasks. Like MADDPG, our approach uses deep deterministic policy gradients [20] to learn decentralised policies. However, FACMAC learns a single centralised but factored critic, which factors the joint action-value function Qtot into per-agent utilities Qa that are combined via a non-linear monotonic function, as in the popular Q-learning algorithm QMIX [31]. While the critic used in COMA and MADDPG is also centralised, it is monolithic rather than factored.4 Compared to learning a monolithic critic, our factored critic can potentially scale better to tasks with a larger number of agents and/or actions. In addition, in contrast to other value-based approaches such as QMIX, there are no inherent constraints on factoring the critic. This allows us to employ rich value factorisations, including nonmonotonic ones, that value-based methods cannot directly use without forfeiting decentralisability or introducing other significant algorithmic changes. We thus also employ a nonmonotonic factorisation and empirically demonstrate that its increased representational capacity allows it to solve some tasks that cannot be solved with monolithic, or monotonically factored critics.

In MADDPG, a separate policy gradient is derived for each agent individually, which optimises its policy assuming all other agents actions are fixed. This could cause the agents to converge to sub-optimal policies in which no single agent wishes to change its action unilaterally. In FACMAC, we use a new centralised gradient estimator that optimises over the entire joint action space, rather than optimising over each agent s action space separately as in MADDPG. The agents policies are thus trained as a single joint-action policy, which can enable learning of more coordinated behaviour, as well as the ability to escape sub-optimal solutions. The centralised gradient estimator fully reaps the benefits of learning a centralised critic, by not implicitly marginalising over the actions of the other agents in the policy-gradient update. The gradient estimator used in MADDPG is also known to be vulnerable to relative overgeneralisation [47]. To overcome this issue, in our centralised gradient estimator, we sample all actions from all agents current policies when evaluating the joint action-value function. We empirically show that MADDPG can quickly get stuck in local optima in a simple continuous matrix game, whereas our centralised gradient estimator finds the optimal policy. While Lyu et al. [24] recently show that merely using a centralised critic (with per-agent gradients that optimise over each agent s actions separately) does not necessarily lead to better coordination between agents, our centralised gradient estimator re-establishes the value of using centralised critics.

Most recent works on continuous MARL focus on evaluating their algorithms on the multi-agent particle environments [23], which feature a simple two-dimensional world with some basic simulated physics. To demonstrate FACMAC s scalability to more complex continuous domains and to stimulate more progress in continuous MARL, we introduce Multi-Agent Mu Jo Co (MAMu Jo Co), a new, comprehensive benchmark suite that allows the study of decentralised continuous control. Based on the popular single-agent Mu Jo Co benchmark [5], MAMu Jo Co features a wide variety of novel robotic control tasks in which multiple agents within a single robot have to solve a task cooperatively.

We evaluate FACMAC on variants of the multi-agent particle environments [23] and our novel MAMu Jo Co benchmark, which both feature continuous action spaces, and the challenging SMAC benchmark [35], which features discrete action spaces. Empirical results demonstrate FACMAC s superior performance over MADDPG and other baselines on all three domains. In particular, FACMAC scales better when the number of agents (and/or actions) and the complexity of the task increases. Results on SMAC show that FACMAC significantly outperforms stochastic DOP [46], which recently claimed to be the first multi-agent actor-critic method to outperform state-of-the-art valued-based methods on SMAC, in all scenarios we tested. Moreover, our ablations and additional experiments demonstrate the advantages of both factoring the critic and using our centralised gradient estimator. We show that, compared to learning a monolithic critic, learning a factored critic can: 1) better take advantage of the centralised gradient estimator to optimise the agent policies when the

In MADDPG, each agent learns its own centralised critic, as it is designed for general multi-agent learning problems, including cooperative, competitive, and mixed settings. 4We use centralised and monolithic critic and monolithic critic interchangeably to refer to the centralised critic used in COMA and MADDPG, and centralised but factored critic and factored critic interchangeably to refer to the critic used in our approach.

number of agents and/or actions is large, and 2) leverage a nonmonotonic factorisation to solve tasks that cannot be solved with monolithic or monotonically factored critics.

2 Background

We consider a fully cooperative multi-agent task in which a team of agents interacts with the same environment to achieve some common goal. It can be modeled as a decentralised partially observable Markov decision process (Dec-POMDP) [28] consisting of a tuple G = N, S, U, P, r, Ω, O, γ . Here N {1, . . . , n} denotes the finite set of agents and s S describes the true state of the environment. At each time step, each agent a N selects a discrete or continuous action ua U, forming a joint action u U U n. This results in a transition to the next state s according to the state transition function P(s |s, u) : S U S [0, 1] and a team reward r(s, u). γ [0, 1) is a discount factor. Due to the partial observability, each agent a N draws an individual partial observation oa Ω from the observation kernel O(s, a). Each agent learns a stochastic policy πa(ua|τa) or a deterministic policy µa(τa), conditioned only on its local action-observation history τa T (Ω U) . The joint stochastic policy π induces a joint action-value function: Qπ(st, ut) = Est+1: ,ut+1: [Rt|st, ut], where Rt = P i=0 γirt+i is the discounted return. Similarly, the joint deterministic policy µ induces a joint action-value function denoted Qµ(st, ut). We adopt the centralised training with decentralised execution (CTDE) paradigm [27, 18], where policy training can exploit extra global information that might be available and has the freedom to share information between agents during training. However, during execution, each agent must act with only access to its own action-observation history.

VDN and QMIX. VDN [41] and QMIX [31] are Q-learning algorithms for cooperative MARL tasks with discrete actions. They both aim to efficiently learn a centralised but factored action-value function Qπ tot, using CTDE. To ensure consistency between the centralised and decentralised policies, VDN and QMIX factor Qπ tot assuming additivity and monotonicity, respectively. More specifically, VDN factors Qπ tot into a sum of the per-agent utilities: Qπ tot(τ, u; φ) = Pn a=1 Qπa a (τa, ua; φa). QMIX, however, represents Qπ tot as a continuous monotonic mixing function of each agent s utilities: Qπ tot(τ, u, s; φ, ψ) = fψ s, Qπ1 1 (τ1, u1; φ1), . . . , Qπn n (τn, un; φn) , where fψ

Qa 0, a N. This is sufficient to ensure that the global arg max performed on Qπ tot yields the same result as a set of individual arg max performed on each Qπa a . Here fψ is approximated by a monotonic mixing network, parameterised by ψ. Monotonicity can be guaranteed by non-negative mixing weights. These weights are generated by separate hypernetworks [12], which condition on the full state s. QMIX is trained end-to-end to minimise the following loss:

L(φ, ψ) = ED h ytot Qπ tot(τ, u, s; φ, ψ) 2i , (1)

where the bootstrapping target ytot = r + γ maxu Qπ tot(τ , u , s ; φ , ψ ). Here r is the global reward, and φ and ψ are parameters of the target Q and mixing network, respectively, as in DQN [26]. The expectation is estimated with a minibatch of transitions sampled from an experience replay buffer D [21]. During execution, each agent selects actions greedily with respect to its own Qπa a .

MADDPG. MADDPG [23] is an extension of DDPG [20] to multi-agent settings. It is an actorcritic, off-policy method that uses the paradigm of CTDE to learn deterministic policies in continuous action spaces. In MADDPG, a separate actor and critic is learned for each agent, such that each agent can have its own arbitrary reward function. It is therefore applicable to either cooperative, competitive, or mixed settings. We assume each agent a has a deterministic policy µa(τa; θa), parameterised by θa (abbreviated as µa), and let µ = {µa(τa; θa)}n a=1 be the set of all agent policies. In MADDPG, a centralised and monolithic critic that estimates the joint action-value function Qµ a (s, u1, . . . , un; φa) is learned for each agent a separately. The critic is said to be centralised as it utilises information only available to it during the centralised training phase, the global state s5 and the actions of all agents, u1, . . . , un, to estimate the joint action-value function Qµ a , which is parameterised by φa. This joint action-value function is trained by minimising the following loss:

L(φa) = ED h ya Qµ a (s, u1, . . . , un; φa) 2i , (2)

5If the global state s is not available, the centralised and monolithic critic can condition on the joint observations or action-observation histories.

where ya = ra + γQµ a (s , u 1, . . . , u n|u a=µa(τ a;θ a ); φ a ). Here ra is the reward received by each agent a, u 1, . . . , u n is the set of target policies with delayed parameters θ a , and φ a are the parameters of the target critic. The replay buffer D contains the transition tuples (s, s , u1, . . . , un, r1, . . . , rn).

The following policy gradient can be calculated individually to update the policy of each agent a:

θa J(µa) = ED h θaµa(τa) ua Qµ a (s, u1, . . . , un) ua=µa(τa)

where the current agent a s action ua is sampled from its current policy µa when evaluating the joint action-value function Qµ a , while all other agents actions are sampled from the replay buffer D.

In this section, we propose a new approach called FACtored Multi-Agent Centralised policy gradients (FACMAC) that uses a centralised but factored critic and a centralised gradient estimator to learn continuous cooperative tasks. We start by describing the idea of learning a centralised but factored critic. We then discuss our new centralised gradient estimator and demonstrate its benefit in a simple continuous matrix game. Finally, we discuss how we adapt our method to discrete cooperative tasks.

3.1 Learning a Centralised but Factored Critic

Learning a centralised and monolithic critic conditioning on the global state and the joint action can be difficult and/or impractical when the number of agents and/or actions is large [13]. We thus employ value function factorisation in the multi-agent actor-critic framework to enable scalable learning of a centralised critic in Dec-POMDPs. Another key advantage of adopting value factorisation in an actor-critic framework is that, compared to value-based methods, it allows for a more flexible factorisation as the critic s design is not constrained. One can employ any type of factorisation, including nonmonotonic factorisations that value-based methods cannot directly use without forfeiting decentralisability or introducing other significant algorithmic changes.

Specifically, in FACMAC, all agents share a centralised critic Qµ tot that is factored as:

Qµ tot(τ, u, s; φ, ψ) = gψ s, {Qµa a (τa, ua; φa)}n a=1 , (4)

where φ and φa are parameters of the joint action-value function Qµ tot and agent-wise utilities Qµa a , respectively. In our canonical implementation which we refer to as FACMAC, gψ is a non-linear monotonic function parametrised as a mixing network with parameters ψ, as in QMIX [31]. To evaluate the policy, the centralised but factored critic is trained by minimising the following loss:

L(φ, ψ) = ED h ytot Qµ tot(τ, u, s; φ, ψ) 2 i , (5)

where ytot = r + γQµ tot(τ , µ(τ ; θ ), s ; φ , ψ ). Here D is the replay buffer, and θ , φ , and ψ are parameters of the target actors, critic, and mixing network, respectively.

Leveraging the flexibility of our approach, namely the lack of restrictions on the form of the critic, we also explore a new nonmonotonic factorisation with full representational capacity. The joint action-value function Qµ tot is represented as a non-linear nonmonotonic mixing function of per-agent utilities Qµa a . This nonmonotonic mixing function is parameterised as a mixing network, with a similar architecture to gψ in FACMAC, but without the constraint of monotonicity enforced by using non-negative weights. We refer to this method as FACMAC-nonmonotonic. Additionally, to better understand the advantages of factoring a centralised critic, we also explore two additional simpler factorisation schemes. These include factoring the centralised critic Qµ tot into a sum of per-agent utilities Qµa a as in VDN (FACMAC-vdn), and as a sum of Qµa a and a state-dependent bias (FACMAC-vdn-s). Our value factorisation technique is general and can be readily applied to any multi-agent actor-critic algorithms that learn centralised and monolithic critics [23, 9, 8].

3.2 Centralised Policy Gradients

To update the decentralised policy of each agent, a naive adaptation of the deterministic policy gradient used by MADDPG (shown in (3)) is

θa J(µa) = ED h θaµa(τa) ua Qµ tot(τ, u1, . . . , un, s) ua=µa(τa)

Figure 1: The overall FACMAC architecture. (a) The decentralised policy networks. There is a sampling step since we sample from the categorical distribution when using discrete actions. (b) The centralised but factored critic. (c) The non-linear monotonic mixing function.

Compared to the policy gradient used in MADDPG, the updates of all agents individual deterministic policies now depend on the single shared factored critic Qµ tot, as opposed to learning and utilising a monolithic critic Qµ a for each agent. However, there are two main problems in both policy gradients. First, each agent optimises its own policy assuming all other agents actions are fixed, which could cause the agents to converge to sub-optimal policies in which no single agent wishes to change its action unilaterally. Second, both policy gradients make the corresponding methods vulnerable to relative overgeneralisation [47] as, when agent a ascends the policy gradient based on Qµ a or Qµ tot, only its own action ua is sampled from its current policy µa, while all other agents actions are sampled from the replay buffer D. The other agents actions thus might be drastically different from the actions their current policies would choose. This could cause the agents to converge to sub-optimal actions that appear to be a better choice when considering the effect of potentially arbitrary actions from the other collaborating agents.

In FACMAC, we use a new centralised gradient estimator that optimises over the entire joint action space, rather than optimising over each agent s actions separately as in both (3) and (6), to achieve better coordination among agents. In addition, to overcome relative overgeneralisation, when calculating the policy gradient we sample all actions from all agents current policies when evaluating Qµ tot. Our centralised policy gradient can thus be estimated as

θJ(µ) = ED h θµ µQµ tot(τ, µ1(τ1), . . . , µn(τn), s) i , (7)

where µ = {µ1(τ1; θ1), . . . , µn(τn; θn)} is the set of all agents current policies and all agents share the same actor network parameterised by θ. However, it is not a requirement of our method for all agents to share parameters in this manner.

If the critic factorisation is linear, as in FACMAC-vdn, then the centralised gradient is equivalent to the per-agent gradients that optimise over each agent s actions separately. This is explored in more detail by DOP [46], which restricts the factored critic to be linear to exploit this equivalence. A major benefit of our method then, is that it does not place any such restrictions on the critic. As remarked by Lyu et al. [24], merely using a centralised critic with per-agent gradients does not necessarily lead to better coordination between agents due to the two problems outlined above. Even with a factored critic, methods that use stochastic policy gradients can still suffer from the problems caused by the per-agent gradients if a fully factored stochastic policy is used. Our centralised gradient estimator, which uses deterministic policies and optimises over the entire joint action space, is required in order to fully take advantage of a centralised critic.

Figure 1 illustrates the overall FACMAC architecture. For each agent a, there is one policy network that decides which individual action (discrete or continuous) to take. There is also one critic network for each agent a that estimates the individual agent utilities Qa, which are then combined into the joint action-value function Qtot via a non-linear monotonic mixing function as in QMIX. Qtot is then used by our centralised gradient estimator to help the actor update its policy parameters.

To show the benefits of our new centralised gradient estimator, we compare MADDPG with the centralised policy gradient (CPG) against the original MADDPG on a simple continuous cooperative matrix game with two agents. Figure 10 in Appendix D.1 illustrates this matrix game. There is a

Figure 2: (a) Mean test return on Continuous Matrix Game. (b) Left: Per-agent policy gradient at the origin. For agent 1 (similarly for agent 2) it is 0 since the gradient term assumes the other agent s action to be fixed and thus it only considers the relative improvements along the dotted line (agent 1 s own action space). Right: Our centralised policy gradient correctly determines the gradient for improving the joint action.

narrow path (shown in red) starting from the origin (0, 0) to (1, 1), in which the reward gradually increases. Everywhere else there is a small punishment moving away from the origin, increasing in magnitude further from the origin. Experimental results are shown in Figure 2(a). MADDPG quickly gets stuck in the local optimum within 200k timesteps, while MADDPG (with CPG) robustly converges to the optimal policy. Figure 2(b) visualises the differences between the per-agent and centralised policy gradients, demonstrating that the per-agent policy gradient can be wrong in our continuous matrix game and our centralised policy gradient is necessary to take advantage of the centralised critic. In Section 5, we further demonstrate the benefits of our centralised gradient estimator in more complex tasks.

3.3 Discrete Policy Learning

As FACMAC requires differentiable policies and the sampling process of discrete actions from a categorical distribution is not differentiable, we use the Gumbel-Softmax estimator [14] to enable efficient learning of FACMAC on cooperative tasks with discrete actions. The Gumbel-Softmax estimator is a continuous distribution that approximates discrete samples from a categorical distribution to produce differentiable samples. Moreover, we use the Straight-Through Gumbel-Softmax Estimator [14] to ensure the action dynamics during training and evaluation are the same. Specifically, during training, we sample discrete actions ua from the original categorical distribution in the forward pass, but use the continuous Gumbel-Softmax sample xa in the backward pass to approximate the gradients: θaua θaxa. We can then update the agent s policy using our centralised policy gradient:

θJ(x) ED h θx x Qx tot(τ, x1, . . . , xn, s) i , where x = {x1, . . . , xn} is the set of continuous samples that approximates the discrete agent actions. The softmax temperature hyperparameter τ is set to be 1 in our experiments.

4 Multi-Agent Mu Jo Co

The evaluation of continuous MARL algorithms has recently been largely limited to the simple multi-agent particle environments [23]. We believe the lack of diverse continuous benchmarks is one factor limiting progress in continuous MARL. To demonstrate FACMAC s scalability to more complex continuous domains and to stimulate more progress in continuous MARL, we develop Multi-Agent Mu Jo Co (MAMu Jo Co), a novel benchmark for continuous cooperative multi-agent robotic control. Starting from the popular fully observable single-agent robotic Mu Jo Co [42] control suite included with Open AI Gym [5], we create a wide variety of novel scenarios in which multiple agents within a single robot have to solve a task cooperatively.

This design offers important benefits. It facilitates comparisons to existing literature on both the fully observable single-agent domain [29], as well as settings with low-bandwidth communication [45]. More importantly, it allows for the study of novel MARL algorithms for decentralised coordination in isolation (scenarios with multiple robots may add confounding factors such as spatial exploration), which is currently a gap in the research literature. MAMu Jo Co also includes scenarios with a larger and more flexible number of agents, which takes inspiration from modular robotics [49, 19]. Compared to traditional robots, modular robots are more versatile, configurable, and scalable. We thus

Figure 3: Agent partitionings for MAMu Jo Co environments: A) Manyagent Swimmer, B) 3Agent Hopper [3x1], C) 2-Agent Half Cheetah [2x3], D) 6-Agent Half Cheetah [6x1], E) 2-Agent Humanoid and 2-Agent Humanoid Standup (each [1x9,1x8]), F) 2-Agent Walker [2X3], G) 2-Agent Reacher [2x1], H) 2-Agent Ant [2x4], I) 2-Agent Ant Diag [2x4], J) 4-Agent Ant [4x2], and K) Manyagent Ant. Colours indicate agent partitionings. Each joint corresponds to a single controllable motor. Split partitions indicate shared body segments. Square brackets indicate [(number of agents) x (joints per agent)]. Joint IDs are in order of definition in the corresponding Open AI Gym XML asset files [5]. Global joints indicate degrees of freedom of the center of mass of the composite robotic agent.

develop two scenarios named Many Agent Swimmer and Many Agent Ant, in which one can configure an arbitrarily large number of agents (within the memory limits), each controlling a consecutive segment of arbitrary length.

Single-robot multi-agent tasks in MAMu Jo Co arise by first representing a given single robotic agent as a body graph, where vertices (joints) are connected by adjacent edges (body segments), as shown in Figure 3. We then partition the body graph into disjoint sub-graphs, one for each agent, each of which contains one or more joints that can be controlled. Note that in Many Agent Swimmer (see Figure 3A) and Many Agent Ant (see Figure 3K), the number of agents are not limited by the given single robotic agent. See Appendix B for more details about MAMu Jo Co.

5 Experimental Results

In this section we present our experimental results on our cooperative variants of the continuous simple tag environment introduced by Lowe et al. [23] (we refer to this environment as Continuous Predator-Prey), our novel continuous benchmark MAMu Jo Co, and the challenging SMAC6 [35] benchmark with discrete action spaces. In discrete cooperative tasks, we compare with state-of-the-art multi-agent actor-critic algorithms MADDPG [23], COMA [9], Central V [9], DOP [46], VDAC-mix [40], and value-based methods QMIX [31] and QPLEX [44]. In continuous cooperative tasks, we compare with MADDPG [23] and independent DDPG (IDDPG), as well as COVDN and COMIX, two novel baselines described below. We also explore different forms of critic factorisation to better understand the advantages of factoring a centralised critic. More details about the environments, experimental setup, and training details are included in Appendix D and E. Code is available at https://github.com/oxwhirl/facmac.

COVDN and COMIX. We find that not many multi-agent value-based methods work off the shelf with continuous actions. To compare FACMAC against value-based approaches in continuous cooperative tasks, we use existing continuous Q-learning approaches in single-agent settings to extend VDN and QMIX to continuous action spaces. Specifically, we introduce COVDN and COMIX, which use VDN-style and QMIX-style factorisation respectively and both perform approximate greedy action selection using the cross-entropy method (CEM) [7]. CEM is a sampling-based derivative-free

6We utilise SC2.4.10., which is used by the latest Py MARL [35] framework. The original results reported in Samvelyan et al. [35] and Rashid et al. [33] use SC2.4.6. Performance is not always comparable across versions.

(a) 3 agents and 1 prey

(b) 6 agents and 2 preys

(c) 9 agents and 3 preys

Figure 4: Mean episode return on Continuous Predator-Prey with different number of agents and preys. The mean across 5 seeds is plotted and the 95% confidence interval is shown shaded. The numbers in square brackets in the figure legend represent the number of random seeds used to run each method (similarly for all other figures).

Figure 5: Mean episode return on different MAMu Jo Co tasks. In Many Agent Swimmer, we configure the number of agents to be 10, each controlling a consecutive segment of length 2. The mean across 7 seeds is plotted and the 95% confidence interval is shown shaded.

heuristic search method that has been successfully used to find approximate maxima of nonconvex Q-networks in single-agent robotic control tasks [15]. The centralised but factored Qtot allows us to use CEM to sample actions for each agent independently and to use the per-agent utility Qa to guide the selection of maximal actions. We do not consider COVDN and COMIX significant algorithmic contributions but instead merely baseline algorithms. See Appendix C for more details about them.

FACMAC outperforms MADDPG and other baselines in both discrete and continuous action tasks. Figure 4 and 5 illustrate the mean episode return attained by different methods on Continuous Predator-Prey with varying number of agents and different MAMu Jo Co tasks, respectively. We can see that FACMAC significantly outperforms MADDPG on all these continuous cooperative tasks, both in terms of absolute performance and learning speed. On discrete SMAC tasks, Figure 6 shows that FACMAC performs significantly better than MADDPG on 4 out of 6 maps we tested, and achieves similar performance to MADDPG on the other 2 maps. Additionally, on all 6 SMAC maps, FACMAC significantly outperforms all multi-agent actor-critic baselines (COMA, Central V, DOP, and VDAC-mix), while DOP is recently claimed to be the first multi-agent actor-critic method that outperforms state-of-the-art valued-based methods on SMAC. FACMAC is also competitive with state-of-the-art value-based methods (QMIX and QPLEX), with significantly better performance on MMM, bane_vs_bane, MMM2, and 27m_vs_30m. These results demonstrate the benefits of our method for improving performance in challenging cooperative tasks with discrete and continuous action spaces.

In Continuous Predator-Prey, FACMAC-vdn scales better than FACMAC when the number of agents increases. However, on MAMu Jo Co, FACMAC-vdn performs drastically worse than FACMAC in 2-Agent Humanoid and Many Agent Swimmer (with 10 agents), demonstrating the necessity of the non-linear mixing of agent utilities and conditioning on the central state information in order to achieve competitive performance in such tasks. Furthermore, on SMAC, Figure 13 in Appendix F shows that FACMAC is noticeably more stable than FACMAC-vdn and FACMAC-vdn-s across different maps, and achieves significantly better performance on the super hard map MMM2.

Interestingly, we find that FACMAC performs similarly to COMIX on both Continuous Predator-Prey and MAMu Jo Co tasks. As FACMAC and COMIX use the same value factorisation as in QMIX and are both off-policy, this suggests that, in these continuous cooperative tasks, factorisation of the

Figure 6: Median test win % on six different SMAC maps, including 2s3z (easy), MMM (easy), 2c_vs_64zg (hard), bane_vs_bane (hard), MMM2 (super hard), and 27m_vs_30m (super hard). The median across 5 seeds is plotted and the 25 75% percentiles is shown shaded. The performance of the heuristic-based algorithm is shown as a dashed line. We report the median instead of the mean as recommended by Samvelyan et al. [35] in order to avoid the effect of any outliers.

joint Q-value function plays a greater role in performance than the underlying algorithmic choices. On SMAC, however, FACMAC performs significantly better than QMIX on MMM, bane_vs_bane, MMM2, and 27m_vs_30m. For instance, on bane_vs_bane, a task with 24 agents, while QMIX struggles to find the optimal policy with 2 million timesteps, FACMAC, with exactly the same value factorisation, can quickly recover the optimal policy and achieve 100% test win rate. This shows the convergence advantages of policy gradient methods in this type of multi-agent settings [43].

FACMAC scales better as the number of agents (and/or actions) and the complexity of the task increases. As shown in Figure 4(b) and 4(c), MADDPG performs poorly if we increase the number of agents in Continuous Predator-Prey, while both FACMAC and FACMAC-vdn achieve significantly better performance. The monolithic critic in MADDPG simply concatenates all agents observations into a single input vector, which can be quite large when there are many agents and/or entities and make it more difficult to learn a good critic. Factoring the critic enables scalable critic learning, by combining individual agent utilities that condition on much smaller observations into a joint action-value function. On MAMu Jo Co (shown in Figure 5), similarly, the largest performance gap between FACMAC and MADDPG can be seen on Many Agent Swimmer (with 10 agents), a task with the largest number of agents among three MAMu Jo Co tasks tested.

On SMAC (shown in Figure 6), the largest performance gap between FACMAC and MADDPG can be seen on the challenging MMM2 and 27m_vs_30m with a large number of agents, which are classified as 2 super hard SMAC maps due to current methods poor performance [35]. We can see that FACMAC is able to learn to consistently defeat the enemy, whereas MADDPG fails to learn anything useful in both tasks. The second largest performance gap between FACMAC and MADDPG can be seen on the hard map 2c_vs_64zg, where MADDPG not only performs significantly worse but also exhibits significantly more variance than FACMAC across seeds. While there are only 2 agents in this scenario, the number of actions each agent can choose is the largest among all 6 maps tested as there are 64 enemies. These results further demonstrate that FACMAC scales better when the number of agents (and/or actions) and the complexity of the tasks increases.

Factoring the critic can better take advantage of our centralised gradient estimator to optimise the agent policies when the number of agents and/or actions is large. We conduct ablation experiments to investigate the influence of factoring the critic and using the centralised gradient estimator in our method. FACMAC (without CPG) is our method without the centralised policy gradient. It uses a naive adaptation of the deterministic policy gradient used in MADDPG (shown in (6)). Thus, the only difference between FACMAC (without CPG) and MADDPG is that the previous

one learns a non-linearly factored critic while the latter one learns a monolithic critic. We also evaluate MADDPG with our centralised policy gradient and refer to it as MADDPG (with CPG).

Figure 7: Ablations for different FACMAC components on SMAC and MAMu Jo Co tasks.

Figure 7 shows the ablation results on SMAC and MAMu Jo Co. We can see that FACMAC (without CPG) performs significantly better than MADDPG on both SMAC maps tested, demonstrating the advantages of factoring the critic in challenging coordination problems. With the centralised policy gradient, MADDPG (with CPG) performs significantly better than MADDPG on 2c_vs_64zg. However, on the harder map MMM2, MADDPG with both policy gradients fail to learn anything useful. By contrast, FACMAC significantly outperforms FACMAC (without CPG) on MMM2, and has lower variance across seeds on 2c_vs_64zg. Furthermore, on Many Agent Swimmer with 2 agents, our centralised gradient estimator does not affect the performance of both MADDPG and FACMAC. However, on the same task with 10 agents, using the centralised gradient estimator significantly improves the learning performance when learning a factored critic. These results demonstrate that factoring the critic can better take advantage of our centralised gradient estimator to optimise the agent policies when the number of agents and/or actions is large.

Figure 8: Mean episode return on (Left) Continuous Matrix Game and (Right) a variant of our Continuous Predator-Prey task (with 3 agents and 1 prey) with nonmonotonic value functions.

Nonmonotonically factored critics can solve tasks that cannot be solved with monolithic or monotonically factored critics. In our multi-agent actor-critic framework, there are no inherent constraints on factoring the critic, we thus also employ a nonmonotonic factorisation and refer to it as FACMAC-nonmonotonic (as discussed in Section 3.1). As shown in Figure 8 (left), on our continuous matrix game (as discussed in Section 3.2), FACMAC-nonmonotonic can robustly learn the optimal policy, while both FACMAC and MADDPG converge to some sub-optimal policy within 200k timesteps. On a variant of Continuous Predator-Prey task (see Appendix D.2 for more details about this task) with nonmonotonic value functions (i.e., an agent s ordering over its own actions depends on other agents actions [25]), Figure 8 (right) shows that both FACMAC and MADDPG fail to learn anything useful, while FACMAC-nonmonotonic successfully learns to capture the prey. These results demonstrate that nonmonotonically factored critics can solve tasks that cannot be solved with monolithic or monotonically factored critics.

It is important to note that the relative performance of FACMAC and FACMAC-nonmonotonic is task dependent. On the original Continuous Predator-Prey task (with 3 agents and 1 prey), FACMACnonmonotonic yields similar performance to FACMAC (see Figure 12 in Appendix F). On SMAC (see Figure 13 in Appendix F), FACMAC-nonmonotonic performs similarly to FACMAC on easy maps, but exhibits significantly worse performance on harder maps. This shows that, in this type of tasks, using an unconstrained factored critic could lead to an increase in learning difficulty. We thus expect FACMAC-nonmonotonic to be more useful in tasks with nonmonotonic value functions.

6 Conclusion

This paper presented FACMAC, a multi-agent actor-critic method that learns decentralised policies with a centralised but factored critic, working for both discrete and continuous cooperative tasks. We showed the advantages of both factoring the critic and using the new centralised gradient estimator in our approach. We also introduced a novel benchmark suite MAMu Jo Co to demonstrate FACMAC s scalability to more complex continuous tasks. Our results on three different domains demonstrated FACMAC s superior performance over existing MARL algorithms. Future work will explore more forms of nonmonotonic factorisation to tackle tasks with nonmonotonic value functions.

Acknowledgements

We would like to thank Dr Martin Strohmeier and the members of the Whiteson Research Lab for their helpful feedback. We would also like to thank the anonymous reviewers for their constructive comments during the reviewing process. This project has received funding from the European Research Council (ERC), under the European Union s Horizon 2020 research and innovation programme (grant agreement number 637713). It was also supported by an EPSRC grant (EP/M508111/1, EP/N509711/1). Christian Schroeder de Witt is generously funded by Cyber Defence Campus, Science and Technology, Armasuisse, Switzerland. The experiments were made possible by a generous equipment grant from NVIDIA.

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1. For all authors...

(a) Do the main claims made in the abstract and introduction accurately reflect the paper s contributions and scope? [Yes] (b) Did you describe the limitations of your work? [Yes] We mention that the relative performance of FACMAC and FACMAC-nonmonotonic is task dependent and discuss some empirical results in the last paragraph of Section 5. (c) Did you discuss any potential negative societal impacts of your work? [Yes] See Appendix B.1. (d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] 2. If you are including theoretical results...

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(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We include the code and instructions needed to reproduce the main experimental results in the supplemental material. (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix E. (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] For all results on Continuous Matrix Game, Continuous Predator-Prey, and MAMu Jo Co, we report the mean episode return and the 95% confidence interval. For all results on SMAC, we report the median test win rate as well as the 25-75% percentiles (as in [35]). (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We describe the type of GPUs used in Appendix E. 4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...

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