E VOLUTIONARY P OPULATION C URRICULUM FOR S CALING M ULTI -A GENT R EINFORCEMENT L EARNING
Anonymous authors
Paper under double-blind review
A BSTRACT
In multi-agent games, the complexity of the environment can grow exponentially as the number of agents increases, so it is particularly challenging to learn good policies when the agent population is large.
In this paper, we introduce Evolutionary
Population Curriculum (EPC), a curriculum learning paradigm that scales up MultiAgent Reinforcement Learning (MARL) by progressively increasing the population of training agents in a stage-wise manner.
Furthermore, EPC uses an evolutionary approach to ﬁx an objective misalignment issue throughout the curriculum: agents successfully trained in an early stage with a small population are not necessarily the best candidates for adapting to later stages with scaled populations.
Concretely,
EPC maintains multiple sets of agents in each stage, performs mix-and-match and ﬁne-tuning over these sets and promotes the sets of agents with the best adaptability to the next stage.
We implement EPC on a popular MARL algorithm, MADDPG, and empirically show that our approach consistently outperforms baselines by a large margin as the number of agents grows exponentially.
I NTRODUCTION
Most real-world problems involve interactions between multiple agents and the problem becomes signiﬁcantly harder when there exist complex cooperation and competition among agents.
Inspired by the tremendous success of deep reinforcement learning (RL) in single-agent applications, such as
Atari games (Mnih et al., 2013), robotics manipulation (Levine et al., 2016), and navigation (Zhu et al., 2017; Wu et al., 2018; Yang et al., 2019), it has become a popular trend to apply deep RL techniques into multi-agent applications, including communication (Foerster et al., 2016; Sukhbaatar et al., 2016;
Mordatch & Abbeel, 2018), trafﬁc light control (Wu et al., 2017), physical combats (Bansal et al., 2018), and video games (Liu et al., 2019; OpenAI, 2018).
A fundamental challenge for multi-agent reinforcement learning (MARL) is that, as the number of agents increases, the problem becomes signiﬁcantly more complex and the variance of policy gradients can grow exponentially (Lowe et al., 2017).
Despite the advances on tackling this challenge via actor-critic methods (Lowe et al., 2017; Foerster et al., 2018), which utilize decentralized actors and centralized critics to stabilize training, recent works still scale poorly and are mostly restricted to less than a dozen agents.
However, many real-world applications involve a moderately large population of agents, such as algorithmic trading (Wellman et al., 2005), sport team competition (Hausknecht & Stone, 2015), and humanitarian assistance and disaster response (Meier, 2015), where one agent should collaborate and/or compete with all other agents.
When directly applying the existing MARL algorithms to complex games with a large number of agents, as we will show in Sec.
5.3, the agents may fail to learn good strategies and end up with little interaction with other agents even when collaboration is signiﬁcantly beneﬁcial.
Yang et al.
(2018) proposed a provably-converged meanﬁeld formulation to scale up the actor-critic framework by feeding the state information and the average value of nearby agents’ actions to the critic.
However, this formulation strongly relies on the assumption that the value function for each agent can be well approximated by the mean of local pairwise interactions.
This assumption often does not hold when the interactions between agents become complex, leading to a signiﬁcant drop in the performance.
In this paper, we propose a general learning paradigm called Evolutionary Population Curriculum (EPC), which allows us to scale up the number of agents exponentially.
The core idea of EPC is to progressively increase the population of agents throughout the training process.
Particularly, we divide the learning procedure into multiple stages with increasing number of agents in the environment.
The agents ﬁrst learn to play in simpler scenarios with less agents and then leverage these experiences to gradually adapt to later stages with more agents and ultimately our desired population.
There are two key components in our curriculum learning paradigm.
To process the varying number of agents during the curriculum procedure, the policy/critic needs to be population-invariant .
So, we choose a self-attention (Vaswani et al., 2017) based architecture which can generalize to an arbitrary number of agents with a ﬁxed number of parameters.
More importantly, we introduce an evolutionary selection process, which helps address the misalignment of learning goals across stages and improves the agents’ performance in the target environment.
Intuitively, our within-stage MARL training objective only incentivizes agents to overﬁt a particular population in the current stage.
When moving towards a new stage with a larger population, the successfully trained agents may not adapt well to the scaled environment.
To mitigate this issue, we maintain multiple sets of agents in each stage, evolve them through cross-set mix-and-match and parallel MARL ﬁne-tuning in the scaled environment, and select those with better adaptability to the next stage.
EPC is RL-algorithm agnostic and can be potentially integrated with most existing MARL algorithms.
In this paper, we illustrate the empirical beneﬁts of EPC by implementing it on a popular MARL algorithm, MADDPG (Lowe et al., 2017), and experimenting on three challenging environments, including a predator-prey-style individual survival game, a mixed cooperative-and-competitive battle game, and a fully cooperative food collection game.
We show that EPC outperforms baseline approaches by a large margin on all these environments as the number of agents grows even exponentially.
We also demonstrate that our method can improve the stability of the training procedure.
R ELATED W ORK
Multi-Agent Reinforcement Learning: It has been a long history in applying RL to multi-agent games (Littman, 1994;
Shoham et al., 2003; Panait & Luke, 2005; Wright et al., 2019).
Recently, deep
RL techniques have been applied into the multi-agent scenarios to solve complex Markov games and great algorithmic advances have been achieved.
Foerster et al.
and
He et al.
(2016) explored a multi-agent variant of deep Q-learning; Peng et al.
(2017) studied a fully centralized actor-critic variant; Foerster et al.
(2018) developed a decentralized multi-agent policy gradient algorithm with a centralized baseline; Lowe et al.
(2017) proposes the MADDPG algorithm which extended DDPG to the multi-agent setting with decentralized policies and centralized Q functions.
Our population curriculum approach is a general framework for scaling MARL which can be potentially combined with any of these algorithms.
Particularly, we implement our method on top of the MADDPG algorithm in this paper and take different MADDPG variants as baselines in experiments.
There are also other works studying large-scale MARL recently (Lin et al., 2018; Jiang & Lu, 2018;
Yang et al., 2018; Suarez et al., 2019), which typically simplify the problem by weight sharing and taking only local observations.
We consider a much more general setting with global observations and unsharedweight agents.
Additionally, our approach is a general learning paradigm which is complementary to the speciﬁc techniques proposed in these works.
Attention-Based Policy Architecture: Attention mechanism is widely used in RL policy representation to capture object level information (Duan et al., 2017;
Wang et al., 2018), represent relations (Zambaldi et al., 2018;
Malysheva et al., 2018; Yang et al., 2019)
and extract communication channels (Jiang & Lu, 2018).
Iqbal & Sha (2019) use an attention-based critic.
In our work, we utilize an attention module in both policy and critic, inspired by the transformer architecture (Vaswani et al., 2017), for the purpose of generalization to an arbitrary number of input entities.
Curriculum Learning:
Curriculum learning can be tracked back to Elman (1993), and its core idea is to “start small”: learn the easier aspects of the task ﬁrst and then gradually increase the task difﬁculty.
It has been extended to deep neural networks on both vision and language tasks (Bengio et al., 2009) and much beyond: Karras et al.
(2017) propose to progressively increase the network capacity for synthesizing high quality images; Murali et al.
(2018) apply a curriculum over the control space for robotic manipulation tasks; several works (Wu & Tian, 2016; Florensa et al., 2017;
Sukhbaatar et al., 2017; Wang et al., 2019) have proposed to ﬁrst train RL agents on easier goals and switch to harder ones later.
Baker et al.
(2019) show that multi-agent self-play can also lead to autocurricula in open-ended environments.
In our paper, we propose to progressively increase the number of the agents as a curriculum for better scaling multi-agent reinforcement learning.
Evolutionary Learning: Evolutionary algorithms, originally inspired by Darwin’s natural selection, has a long history (B ¨ ack & Schwefel, 1993), which trains a population of agents in parallel, and let them evolve via crossover, mutation and selection processes.
Recently, evolutionary algorithms have been applied to learn deep RL policies with various aims, such as to enhance training scalability (Salimans et al., 2017), to tune hyper-parameters (Jaderberg et al., 2017), to evolve intrinsic dense rewards (Jaderberg et al., 2018), to learn a neural loss for better generalization (Houthooft et al., 2018), to obtain diverse samples for faster off-policy learning (Khadka & Tumer, 2018), and to encourage exploration (Conti et al., 2018).
Leveraging this insight, we apply evolutionary learning to better scale MARL : we train several groups of agents in each curriculum stage and keep evolving them to larger populations for the purpose of better adaptation towards the desired population scale and improved training stability.
Czarnecki et al.
(2018) proposed a similar evolutionary mix-and-match training paradigm to progressively increase agent capacity, i.e., larger action spaces and more parameters.
Their work considers a ﬁxed environment with an increasingly more complex agent and utilizes the traditional parameter crossover and mutation during evolution.
By contrast, we focus on scaling MARL, namely an increasingly more complex environment with a growing number of agents.
More importantly, we utilize MARL ﬁne-tuning as an implicit mutation operator rather than the classical way of mutating parameters, which is more efﬁcient, guided and applicable to even a very small number of evolution individuals.
A similar idea of using learning for mutation is also considered by Gangwani & Peng (2018) in the single-agent setting.
B ACKGROUND
Markov Games: We consider a multi-agent Markov decision processes (MDPs) (Littman, 1994).
Such an N -agent Markov game is deﬁned by state space S of the game, action spaces A 1 , ..., A N and observation spaces O 1 , ..., O N for each agent.
Each agent i receives a private observation correlated with the state
o i : S
(cid:55)→ O i and produces an action by a stochastic policy πππ θ i : O i × A i
(cid:55)→ [0 , 1] parameterized by θ i .
Then the next states are produced according to the transition function T :
S ×
A 1 × ...
×
A N (cid:55)→ S .
The initial state is determined by a distribution ρ : S (cid:55)→ [0 , 1] .
Each agent i obtains rewards as a function of the state and its action r
i : S ×
A i (cid:55)→ R , and aims to maximize its own expected return
R i =
(cid:80) Tt =
γ t r ti ( s t , a ti ) , where γ is a discount factor and T is the time horizon.
To minimize notation, we omit subscript of policy when there is no ambiguity.
Multi-Agent Deep Deterministic Policy Gradient (MADDPG): MADDPG (Lowe et al., 2017) is a multi-agent variant of the deterministic policy gradient algorithm
(Silver et al., 2014).
It learns a centralized Q function for each agent which conditions on global state information to resolve the non-stationary issue.
Consider N agents with deterministic policies µµµ = { µµµ 1 , ...,µµµ N } where µµµ i :
O i (cid:55)→
A i is parameterized by θ i .
The policy gradient for agent i is:
[Equation]
Here D denotes the replay buffer while Q µµµi ( x , a 1 , ..., a N ) is a centralized action-value function for agent i that takes the actions of all agents, a 1 , . . .
, a N and the state information x (i.e., x = ( o 1 , ..., o N ) or simply x = s if s is available).
Let x (cid:48) denote the next state from the environment transition.
The replay buffer D contains experiences in the form of tuples ( x , x (cid:48) , a 1 , . .
. , a N , r 1 , . .
. , r N ) .
Suppose the centralized critic Q µµµ i is parameterized by φ i .
Then it is updated via:
[Equation]
where µµµ (cid:48) = { µµµ θ (cid:48) 1 , ...,µµµ θ (cid:48) N } is the set of target policies with delayed parameters θ (cid:48)
i .
Note that the centralized critic is only used during training.
At execution time, each policy µµµ θ i remains decentralized and only takes local observation
o i .
E VOLUTIONARY P OPULATION C URRICULUM
In this section, we will ﬁrst describe the base network architecture with the self-attention mechanism (Vaswani et al., 2017) which allows us to incorporate a ﬂexible number of agents during training.
Then we will introduce the population curriculum paradigm and the evolutionary selection process.
4. P OPULATION -I NVARIANT A RCHITECTURE
We describe our choice of architecture based on the MADDPG algorithm (Lowe et al., 2017), which is population-invariant in the sense that both the Q function and the policy can take in an arbitrary number of input entities.
We ﬁrst introduce the Q function (Fig. 1) and then the policy.
FC ! " ! " ! " ! " …
Attention # "
FC
Value $ " ,& " $ ' ,& ' $ ( ,& ( $ ) ,& ) * "+ (a)
FC …
Attention
FC $ ,,, $ ,,' $ ,,( $ ,,-.'
(b)
Attention ! "
($ , , & , )
FC $ ,,- & ,
We adopt the decentralized execution framework, so each agent has its own Q function and policy network.
Particularly for agent i , its centralized Q function is represented as follows:
[Equation]
v i
[Equation]
attention [Equation] Here f i ( o j , a j ) is an observation-action encoder (the green box in Fig.
1(a)) which takes in the observation o j and the action a j from agent j , and outputs the agent embedding of agent j ; v i denotes the global attention embedding (the orange box in Fig. 1(a)) over all the agent embeddings.
We will explain v i and f i later.
g i is a 1-layer fully connected network processing the embedding of the i th agent’s own observation and action.
h i is a 2-layer fully connected network that takes the concatenation of the output of g i and the global attention embedding v i and outputs the ﬁnal Q value.
Attention Embedding v i : We deﬁne the attention embedding v i by a weighted sum of each agent’s embedding f i ( o j , a j ) for j (cid:54)
[Equation]
The coefﬁcient α i,j is computed by
[Equation]
where W ψ and W φ are parameters to learn.
β i,j computes the correlation between the embeddings of agent i and every other agent j via an inner product.
α i,j is then obtained by normalizing β i,j by a softmax function.
Since we represent the observations and actions of other agents with a weighted mean v i from Eq.
4, we can model the interactions between agent i and an arbitrary number of other agents, which allows us to easily increase the number of agents in our curriculum training paradigm.
Observation-Action Encoder f i : We now deﬁne the structure of f i ( o j , a j ) (Fig. 1(b)).
Note that the observation of agent j , o j , also includes many entities, i.e., states of all visible agents and objects in the game.
Suppose o j contains M entities, i.e., o j
= [ o j, 1 , . .
, o j,M ] .
M may also vary as the agent population scales over training procedure or simply during an episode when some agents die.
Thus, we apply another attention module to combine these entity observations together in a similar way to how v i is computed (Eq. 4, 5).
In more details, we ﬁrst apply an entity encoder for each entity type to obtain entity embeddings of all the entities within that type.
For example, in o j , we can have embeddings for agent entities (green boxes in Fig. 1(b)) and landmark/object entities (purple boxes in Fig. 1(b)).
Then we apply an attention module over each entity type by attending the entity embedding of agent j to all the entities of this type to obtain an attended type embedding (the orange box in Fig. 1(b)).
Next, we concatenate all the type embeddings together with the entity embedding of agent j as well as its action embedding.
Finally, this concatenated vector is forwarded to a fully connected layer to generate the output of f i ( o j , a j ) .
Note that in the overall critic network of agent i , the same encoder f
i is applied to every observation-action pair so that the network can maintain a ﬁxed size of parameters even when the number of agents increases signiﬁcantly.
Policy Network: The policy network µµµ i
( o i ) has a similar structure as the observation-action encoder
f i ( o i , a i ) , which uses an attention module over the entities of each type in the observation o i to adapt to the changing population during training.
The only difference in this network is that the action a i is not included in the input.
Notably, we do not share parameters between the Q function and the policy.
4. P OPULATION C URRICULUM
We propose to progressively scale the number of agents in MARL with a curriculum.
Before combining with the evolutionary selection process, we ﬁrst introduce a simpler version, the vanilla population curriculum (PC), where we perform the following stage-wise procedure: (i) the initial stage starts with MARL training over a small number of agents using MADDPG and our populationinvariant architecture; (ii) we start a new stage and double 1 the number of agents by cloning each of the existing agents; (iii) apply MADDPG training on this scaled population until convergence; (iv) if the desired number of agents is not reached, go back to step (ii).
Mathematically, given N trained agents with parameters θθθ = { θ 1 , ..., θ N } from the previous stage, we want to increase the number of the agents to 2 N with new parameters ˜ θθθ
= { ˜ θ 1 , ..., ˜ θ N , ..., ˜ θ 2 N } for the next stage .
In this vanilla version of population curriculum, we simply initialize ˜ θθθ by setting ˜ θ
i ← θ i and ˜ θ N + i ← θ i , and then continue MADDPG training on ˜ θ θ θ to get the ﬁnal policies for the new stage.
Although ˜ θ i and ˜ θ N + i are both initialized from θ i , as training proceeds, they will converge to different policies since these policies are trained in a decentralized manner in MADDPG.
4. E VOLUTIONARY S ELECTION
Introducing new agents by directly cloning existing ones from the previous stage has a clear limitation: the policy parameters suitable for the previous environment are not necessarily the best initialization for the current stage as the population is scaled up.
In the purpose of better performance in the ﬁnal game with our desired population, we need to promote agents with better adaptation abilities during early stages of training.
Therefore, we propose an evolutionary selection process to facilitate the agents’ scaling adaption ability during the curriculum procedure.
Instead of training a single set of agents, we maintain K parallel sets of agents in each stage, and perform crossover, mutation and selection among them for the next stage.
This is the last piece in our proposed Evolutionary Population Curriculum (EPC) paradigm, which is essentially population curriculum enhanced by the evolutionary selection process.
Speciﬁcally, we assume the agents in the multi-agent game have Ω different roles.
Agents in the same role have the same action set and reward structure.
For example, we have Ω = 2 roles in a predator-prey game, namely predators and prey, and Ω = 1 role of agents for a fully cooperative game with homogeneous agents.
For notation conciseness, we assume there are N 1 agents of role 1 , namely A 1 = { µµµ 1 , ...,µµµ N 1 } ; N 2 agents of role 2 , namely A 2 = { µµµ N 1 +1 , ...,µµµ N 1 + N 2 } , and so on.
In each stage, we keep K parallel sets for each role of agents, denoted by A (1) i , . . .
, A ( K ) i for role i , and take a 3-step procedure, i.e., mix-and-match (crossover) , MARL ﬁne-tuning (mutation) and selection , as follows to evolve these K parallel sets of agents for the next stage.
Mix-and-Match (Crossover): In the beginning of a curriculum stage, we scale the population of agents from N to 2 N .
Note that we have K parallel agent sets of size N i for role i , namely
A (1) i , . . .
, A ( K ) i .
We ﬁrst perform a mix-and-match over these parallel sets within every role i : for each set A ( j ) i , we pair it with all the K sets of the same role, which leads to K ( K + 1) / 2 new scaled agent sets of size 2 N i .
Given these scaled sets of agents, we then perform another mix-and-match across all the Ω roles: we pick one scaled set for each role and combine these Ω selected sets to produce a scaled game with 2 N agents.
For example, in the case of Ω = 2 , we can pick one agent set A ( k 1 )from the ﬁrst role and another agent set A ( k 2 )from the second role to form a scaled game.
Thus, there are C max =
( K ( K + 1) / 2)
Ω different combinations in total through this mix-and-match process.
We sample C games from these combinations for mutation in the next step.
Since we are mixing parallel sets of agents, this process can be considered as the crossover operator in standard evolutionary algorithms.
MARL Fine-Tuning (Mutation): In standard evolutionary algorithms, mutations are directly performed on the parameters, which is inefﬁcient in high-dimensional spaces and typically requires a large amount of mutants to achieve sufﬁcient diversity for evolution.
Instead, here we adopt MARL ﬁne-tuning in each curriculum stage (step (iii) in vanilla PC) as our guided mutation operator, which naturally and efﬁciently explores effective directions in the parameter space.
Meanwhile, due to the training variance, MARL also introduces randomness which beneﬁts the overall diversity of the evolutionary process.
Concretely, we apply parallel MADDPG training on each of the C scaled games generated from the mix-and-match step and obtain C mutated sets of agents for each role.
Selection:
Among these C mutated sets of agents for each role, only the best K mutants can survive.
In the case of Ω = 1 , the ﬁtness score of a set of agents is computed as their average reward after
MARL training.
In other cases when Ω ≥ 2 , given a particular mutated set of agents of a speciﬁc role, we randomly generate games for this set of agents and other mutated sets from different agent roles.
We take its average reward from these randomly generated games as the ﬁtness score for this mutated set.
We pick the top- K scored sets of agents in each role to advance to the next curriculum stage.
Algorithm 1: Evolutionary Population Curriculum
Data: environment E ( N, { A i } 1 ≤ i ≤ Ω ) with N agents of Ω roles, desired population N d , initial population N 0 , evolution size K , mix-and-match size C
Result: a set of N d best policies
N ← N 0 ; initialize K parallel agent sets A (1) i , . . .
, A ( K ) i for each role 1 ≤ i ≤ Ω ; initial parallel MARL training on K games, E ( N, { A ( j ) i } 1 ≤ i ≤ Ω ) for 1 ≤ j ≤ K ; while N < N d do
N ← 2 × N ; for 1 ≤ j ≤ C do for each role 1 ≤ i ≤ Ω : j 1 , j 2 ← unif (1 , K ) , ˜ A [Equation] [Equation]
Overall Algorithm:
Finally, when the desired population is achieved, we take the best set of agents in each role based on their last ﬁtness scores as the output.
We conclude the detailed steps of EPC in
Alg. 1.
Note that in the ﬁrst curriculum stage, we just train K parallel games without mix-and-match or mutation.
So, EPC simply selects the best from the K initial sets in the ﬁrst stage while the evolutionary selection process only takes effect starting from the second stage.
We emphasize that although we evolve multiple sets of agents in each stage, the three operators, mix-and-match, MARL ﬁne-tuning and selection, are all perfectly parallel .
Thus, the evolutionary selection process only introduces little inﬂuence on the overall training time.
Lastly, EPC is an RL-algorithm-agnostic learning paradigm that can be potentially integrated with any MARL algorithm other than MADDPG.
E XPERIMENT
We experiment on three challenging environments, including a predatory-prey-style Grassland game, a mixed-cooperative-and-competitive Adversarial Battle game and a fully cooperative Food Collection game.
We compare EPC with multiple baseline methods on these environments with different scales of agent populations and show consistently large gains over the baselines.
In the following, we will ﬁrst introduce the environments and the baselines, and then both qualitative and quantitative performances of different methods on all three environments.
5. E NVIRONMENTS
All these environments are built on top of the particle-world environment (Mordatch & Abbeel, 2018) where agents take actions in discrete timesteps in a continous 2D world.
Grassland :
In this game, we have Ω = 2 roles of agents, N S sheep and N W wolves, where sheep moves twice as fast as wolves.
We also have a ﬁxed amount of L grass pellets (food for sheep) as green landmarks (Fig. 2a).
A wolf will be rewarded when it collides with (eats) a sheep, and the (eaten) sheep will obtain a negative reward and becomes inactive (dead).
A sheep will be rewarded when it comes across a grass pellet and the grass will be collected and respawned in another random position.
Note that in this survival game, each individual agent has its own reward and does not share rewards with others.
Adversarial Battle : This scenario consists of L units of resources as green landmarks and two teams of agents (i.e., Ω = 2 for each team) competing for the resources (Fig. 2b).
Both teams have the same number of agents ( N 1 = N 2 ).
When an agent collects a unit of resource, the resource will be respawned and all the agents in its team will receive a positive reward.
Furthermore, if there are more than two agents from team 1 collide with one agent from team 2, the whole team 1 will be rewarded while the trapped agent from team 2 will be deactivated (dead) and the whole team 2 will be penalized, and vice versa.
Food Collection :
This game has N food locations and N fully cooperative agents ( Ω = 1 ).
The agents need to collaboratively occupy as many food locations as possible within the game horizon (Fig. 2c).
Whenever a food is occupied by any agent, the whole team will get a reward of 6 /N in that timestep for that food.
The more food occupied, the more rewards the team will collect.
In addition, we introduce collision penalties as well as auxiliary shaped rewards for each agent in each game for easier training.
All the environments are fully observable so that each agent needs to handle a lot of entities and react w.r.t.
the global state.
More environment details are in Appx.
A.
M ETHODS AND M ETRIC
We evaluate the following approaches in our experiments: (1) the MADDPG algorithm (Lowe et al., 2017) with its original architecture ( MADDPG ); (2) the provably-converged mean-ﬁeld algorithm (Yang et al., 2018) ( mean-ﬁeld ); (3) the MADDPG algorithm with our population-invariant architecture ( Att-MADDPG ); (4) the vanilla population curriculum without evolutionary selection ( vanilla-PC ); and (5) our proposed EPC approach ( EPC ).
For EPC parameters, we choose K =for Grassland and Adversarial Battle and K = 3 for Food Collection ; for the mix-and-match size C , we simply set it C max and enumerate all possible mix-and-match combinations instead of random sampling. All the baseline methods are trained until the same amount of accumulative episodes as EPC took.
More training details can be found in Appx.
B.
For Grassland and Adversarial Battle with Ω = 2 , we evaluate the performance of different methods by competing their trained agents against our EPC trained agents.
Speciﬁcally, in Grassland , we let sheep trained by each approach compete with the wolves from EPC and collect the average sheep reward as the evaluation metric for sheep.
Similarly, we take the same measurement for wolves from each method.
In Adversarial Battle , since two teams are symmetric, we just evaluate the shared reward of one team trained by each baseline against another team by EPC as the metric.
For Food
Collection with Ω = 1 , since it is fully cooperative, we take the team reward for each method as the evaluation metric.
In addition, for better visualization, we plot the normalized scores by normalizing the rewards of different methods between 0 and 1 in each scale for each game.
More evaluation details are in Appx.
C.
5. Q UALITATIVE R ESULTS
We qualitatively illustrate the learned strategies by MADDPG and EPC in the three games here.
In Grassland , as the number of wolves goes up, it becomes increasingly more challenging for sheep to survive; meanwhile, as the sheep become more intelligent, the wolves will be incentivized to be more aggressive accordingly.
In Fig.
3, we illustrate two representative matches for competition, including one using the MADDPG sheep against the EPC wolves (Fig. 3a), and the other between the
EPC sheep and the MADDPG wolves (Fig. 3b).
From Fig.
3a, we can observe that the MADDPG sheep can be easily eaten up by the EPC wolves (note that dark circle means the sheep is eaten).
On
[Figure] [Figure]
[Figure]
the other hand, in Fig.
3b, we can see that the EPC sheep learns to eat the grass and avoid the wolves at the same time.
In Adversarial Battle , we visualize two matches in Fig.
4 with one over agents by EPC (Fig. 4a) and the other over agents by MADDPG (Fig. 4b).
We can clearly see the collaborations between the EPC agents: although the agents are initially spread over the environment, they learn to quickly gather as a group to protect themselves from being killed.
While for the MADDPG agents, their behavior shows little incentives to cooperate or compete — these agents stay in their local regions throughout the episode and only collect resources or kill enemies very infrequently.
In Food Collection (Fig. 5), the EPC agents in Fig.
5a learn to spread out and occupy as many food as possible to maximize the team rewards.
While only one agent among the MADDPG agents in Fig.
5b successfully occupies a food in the episode.
5. Q UANTITATIVE R ESULTS Quantitative Results in Grassland
In the Grassland game, we perform curriculum training by starting with 3 sheep and 2 wolves, and gradually increase the population of agents.
We denote a game with N S sheep and N W wolves by “scale N S - N W ”.
We start with scale 3 - 2 and gradually increase the game size to scales 6 - 4 , 12and ﬁnally 24 - 16 .For
the two curriculum learning approach, vanilla-PC and EPC, we train over 10episodes in the ﬁrst curriculum stage (scale 3 - 2 ) and ﬁne-tune the agents with 5 × 10 4 episodes after mix-and-match in each of the following stage.
For other methods that train the agents from scratch, we take the same accumulative training iterations as the curriculum methods for a fair comparison.
Main Results: We report the performance of different methods for each game scale in Fig.
6a.
Overall, there are little differences between the mean-ﬁeld approach and the original MADDPG algorithm while the using the population-invariant architecture (i.e., Att-MADDPG) generally boosts the performance of MADDPG.
For the method with population curriculum, vanilla-PC performs almost the same as training from scratch (Att-MADDPG) when the number of agents in the environment is small (i.e., 6 - 4 ) but the performance gap becomes much more signiﬁcant when the population further grows (i.e., 12 - 18 and 24 - 16 ).
For our proposed EPC method, it consistently outperforms all the baselines across all the scales.
Particularly, in the largest scale 24 - 16 , EPC sheep receive 10x more rewards than the best baseline sheep without curriculum training.
Detailed Statistics: Besides rewards, we also compute the statistics of sheep to understand how the trained sheep behave in the game.
We perform competitions between sheep trained by different methods against the EPC wolves and measure the average number of total grass pellets eaten per episode, i.e, #grass eaten , and the average percentage of sheep that survive until the end of an episode, i.e., survival rate , in Fig.
6b.
We can observe that as the population increases, it becomes increasingly harder for sheep to survive while EPC trained sheep remain a high survival rate even on the largest scale.
Moreover, as more sheep in the game, EPC trained sheep consistently learn to eat more grass even under the strong pressure from wolves.
In contrast, the amount of eaten grass of MADDPG approach (i.e., Att-MADDPG) drastically decreases when the number of wolves becomes large.
[Figure]
Quantitative Results in Adversarial Battle
In this game, we evaluate on environments with different sizes of agent population N , denoted by scale N 1 - N 2 where N 1 = N 2 =
N/ 2 .
We start the curriculum from scale 4 - 4 and the increase the population size to scale 8 - 8 ( N = 16 ) and ﬁnally 16 - 16 N = 32 ).
Both vanilla-PC and EPC take 5 × 10 4 training episodes in the ﬁrst stage and then 2 × 10 4 episodes in the following two curriculum stages.
We report the normalized scores of different methods in
Fig.
7, where agents trained by EPC outperforms all the baseline methods increasingly more signiﬁcant as the agent population grows.
Quantitative Results in Food Collection
In this game, we begin curriculum training with N = 3 , namely 3 agents and 3 food locations, and progressively increase the population size N to 6 , 12 and ﬁnally 24 .
Both vanilla-PC and EPC perform training on 5 × 10 4 episodes on the ﬁrst stage of N =
and then 2 × 10 4 episodes in each of the following curriculum stage.
We report the normalized scores for all the methods in Fig.
8, where
EPC is always the best among all the approaches with a clear margin.
Note that the performance of the original MADDPG and the meanﬁeld approach drops drastically as the population size N increases.
Particularly, the mean-ﬁeld approach performs even worse than the original MADDPG method.
We believe this is because in this game, the agents must act according to the global team state collaboratively, which means the local approximation assumption in the mean-ﬁeld approach does not hold clearly.
Ablative Analysis
Stability Analysis: The evolutionary selection process in EPC not only leads to better ﬁnal performances but also stabilizes the training procedure.
We validate the stability of EPC by computing the variance over 3 training seeds for the same experiment and comparing with the variance of vanilla-PC, which is also obtained from 3 training seeds.
Speciﬁcally, we pick the second stage of curriculum learning and visualize the variance of agent scores throughout the stage of training.
These scores are computed by competing against the ﬁnal policy trained by EPC.
We perform analysis on all theenvironments: Grassland with scale 6 - 4 (Fig. 9a), Adversarial Battle with scale 8 - 8 (Fig. 9b) and
Food Collection with scale 6 (Fig. 9c).
We can observe that the variance of EPC is much smaller than vanilla-PC in different games.
Convergence Analysis: To illustrate that the self-attention based policies trained from a smaller scale is able to well adapt to a larger scale via ﬁne-tuning, we pick a particular mutant by EPC in the second curriculum stage and visualize its learning curve throughout ﬁne-tuning for all the environments, Grassland (Fig.
9d), Adversarial Battle (Fig. 9e) and Food Collection (Fig. 9f).
The scores are computed in the same way as the stability analysis.
By comparing to MADDPG and
Att-MADDPG, which train policies from scratch, we can see that EPC starts learning with a much
[Figure] [Figure]
higher score, continues to improve during ﬁne-tuning and quickly converges to a better solution. Note that all baselines are in fact trained much longer. The full convergence curves are in App.
D.1. Generalization: We investigate whether the learned policies can generalize to a different test environment with even a larger scale than the training ones.
To do so, we take the best polices trained by different methods on the largest population and directly apply these policies to a new environment with a doubled population by self-cloning.
We evaluate in all the environments with EPC, vanilla-PC and Att-MADDPG and measure the normalized scores of different methods, which is computed in the same way as the ﬁtness score.
In all cases, we observe a large advantage of EPC over the other two methods, indicating the better generalization ability for policies trained by EPC.
C ONCLUSION
In this paper, we propose to scale multi-agent reinforcement learning by using curriculum learning over the agent population with evolutionary selection.
Our approach has shown signiﬁcant improvements over baselines not only in the performance but also the training stability.
Given these encouraging results on different environments, we believe our method is general and can potentially beneﬁt scaling other MARL algorithms.
We also hope that learning with a large population of agents can also lead to the emergence of swarm intelligence in environments with simple rules in the future.
A E NVIRONMENT D ETAILS
In the Grassland game, sheep gets + 2 reward when he eats the grass, - 5 reward when eaten by wolf.
The wolf get + 5 reward when eats a sheep.
We also shape the reward by distance, sheep will get less negative reward when it is closer to grass and wolf will get less negative reward when it is closer to sheep.
This game is adapted from the original Predator-prey game in the MADDPG paper (Lowe et al., 2017) by introducing grass and allowing agent to die.
In the Adversarial Battle game, agent will get +1 reward when he eats the food, − 6 reward when killed by other agents.
If N agents kill an enemy, they will be rewarded +6 /N .
We shape the reward by distance.
Agent will receive less negative rewards when it is closer to other agents and grass.
We want to encourage collision within agents and also will be easier for them to learn to eat.
This game is adapted from the mean-ﬁeld MARL paper (Yang et al., 2018) by converting it from a grid world to particle-world, introducing food and only allowing 2-agent cooperative killing.
In the Food Collection game, there are N agents and N food locations.
Each agent will get a shared +6 /N reward per timestep when one food is occupied by any agent.
If one agent gets collision with another, all of the agents will get a punish of − 6 /N .
We shape the reward by distance.
Agents will receive less negative rewards when it gets closer to the food.
Since the number of agents and food are equal, we want to avoid the collision within agents and let the agents to learn to occupy as many food as possible.
This is exactly the same game as the Cooperative Navigation game in the MADDPG paper.
We slightly change the reward function to ensure it is bounded w.r.t.
arbitrary number of agents.
We use the normalized reward as the score during evaluation.
For a particular game with a particular scale, we ﬁrst collect the reward for each type of agents, namely the average reward of each individual of that type without the shaped rewards.
Then we re-scale the collected rewards by considering the lowest reward among all methods as score 0 and highest reward as score 1.
B T RAINING D ETAILS
We follow all the hyper-parameters in the original MADDPG paper (Lowe et al., 2017) for both EPC and all the baseline methods considered.
Particularly, we use the Adam optimizer with learning rate 0.01, β 1 = 0 .
9 , β 2 = 0 .
999 and ε = 10 − 8 across all experiments.
τ = 0 .
01 is set for target network update and γ = 0 .
95 is used as discount factor.
We also use a replay buffer of size 10 6 and we update the network parameters after every 100 samples.
The batch size is 1024 .
All the baseline methods are trained for a number of episodes that equals the accumulative number of episodes that EPC has taken.
We set K = 2 in all the games during training except that K = 3 in the food collection game due to computational constraints.
During EPC training, in the grassland game, we train the scale of 3 sheep 2 wolf for 100000 episodes.
We train another 50000 episodes every time the agents number doubles.
In the adversarial battle game and food collection game, we train the ﬁrst scale for 50000 episodes.
We train another 20000 episodes every time the agents number doubles.
In the grassland game, the entity types are the agent itself, other sheep, other wolf and food.
We thus have four types of entity encoders for each of those entity types.
In the adversarial battle game,
Similar to grassland game, the entity types are agent itself, other teammates, enemies and food.
We also have four types of entity encoders for each of those entity types.
Since there is only one group in the food collection game, the entity types are agent itself, other teammates and food.
We thus have three entity encoders in our network.
C E VALUATION D ETAILS
To evaluate the agents trained in the environment with Ω = 2 , we make two roles of agents trained with different approaches compete against each other.
Each competition is simulated forepisodes.
The average normalized reward over the 10000 episodes will be used as the competition score for each side.
Note that in our experiments, we let all the methods compete against our EPC approach for evaluation.
For adversarial battle game, we take the average score of two teams as the model’s ﬁnal evaluation score,
In the food collection game, since there is only one role, we simply simulate the model forepisodes.
The average normalized reward over the 10000 episodes will be used as the score of the model.
D A DDITIONAL D ETAILS ON E XPERIMENT R ESULTS D. F ULL TRAINING CURVES FOR BASELINES
All the baseline methods are trained for a number of episodes that equals the accumulative number of episodes that EPC has taken.
The purpose of Figure 9e,9e,9f is simply showing the transfer performance, i.e., the initialization produced by EPC from the previous stage is effective and can indeed leverage past experiences to warm-start.
The x-axis of the plot was shrunk for visualization purpose.
Here we illustrate the complete convergence curve of baselines, i.e., ATT-MADDPG and MADDPG, in Figure 11a,11b,11c for the 3 games respective.
Although Att-MADDPG takes a much longer time to learn, its performance is still far worse than EPC.
[Figure]
D. R AW REWARD NUMBERS OF EVALUATION RESULTS
In this section, we provide the actual rewards without normalization when comparing all the baselines with EPC.
These scores are corresponding to the histograms reported in Figure 6a, 7, 8.
Grassland game, wolf rewards, corresponding to wolf in Figure 6a
[Table]
Grassland game, sheep rewards, corresponding to sheep in Figure 6a
[Table]
Adversarial Battle game, rewards of team 1, corresponding to Figure
[Table]
Food Collection game, team rewards, corresponding to Figure
[Table]
D. P AIRWISE COMPETITION RESULTS BETWEEN ALL METHODS IN COMPETITIVE GAMES
For visualization purpose, we only illustrate the scores of the competitions between baselines and
EPC in the main paper.
Here we provide the complete competition rewards between every pair of methods in both Grassland and Adversarial Battle with the largest population of agents as follows.
Here show the wolf rewards in Grassland with scale 24-16.
For wolves trained by each approach, we compare them against the sheep by all the methods.
EPC wolves always have the highest rewards as in the bottom row.
Correspondingly, when different wolves compete against EPC sheep, they always obtain the lowest rewards as in the rightmost column.
[Table]
mean
[Table]
Att
[Table]
MADDPG
Vanilla-PC
EPC
MADDPG 66.
mean
[Table]
Att-MADDPG 103.
Vanilla-PC 110.
EPC 120.
Here show the sheep rewards in Grassland with scale 24-16.
For sheep trained by each approach, we compete them against the all different wolves.
EPC sheep always have the highest rewards as in the last row.
Correspondingly, when competing different sheep against EPC wolves, the rewards are always the lowest as in the rightmost column.
[Table]
wolf
[Table]
Here we show the rewards of team 1 in Adversarial Battle with scale 16-16.
For agents trained by each approach, we compare them as team 1 against all different methods as team 2.
When EPC agents as team 1, no matter which opponent is, they always get the highest rewards as in the last row.
When other methods compete against EPC, the obtained rewards are always the lowest as in the rightmost column.
reported
[Table]
D. V ARIANCE OF P ERFORMANCE E VALUATIONS
We present the performance of all approaches in the Food Collection game with the largest scale of 24 agents.
We train all the approaches with 3 different seeds and show the normalized scores with variance as following.
We can see that EPC not only gives better results but also much smaller variance.
E A DDITIONAL E XPERIMENTS ON THE O RIGINAL P REDATOR -P REY G AME
Grassland is adapted from the Predator-prey game introduced by the original MADDPG paper (Lowe et al., 2017).
To further validate our empirical results, we additionally study the performances of different algorithms on the unmodiﬁed Predator-prey game as follows.
We ﬁrst report the normalized score in Fig.
13 by comparing all the methods against EPC.
EPC is consistently better than all methods in all the scales.
in the original Predator-prey game
Besides, we also report the raw reward numbers when competing against EPC.
Since the Predatorprey game is a zero-sum game, we simply report the predator rewards (the prey reward is exactly the negative value).
[Table]
Furthermore
[Table]
we also demonstrate the results of full pairwise competition between every two methods for scale 12-8 below.
Consistently, we can see that EPC predators always have the highest scores as in the last row.
When competing against EPC prey, the lowest rewards are observed.
[Table]