Why do Coverage Gradient Strategies work so nicely in Cooperative MARL? Proof from Coverage Illustration

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In cooperative multi-agent reinforcement studying (MARL), resulting from its on-policy nature, coverage gradient (PG) strategies are usually believed to be much less pattern environment friendly than worth decomposition (VD) strategies, that are off-policy. Nonetheless, some latest empirical research display that with correct enter illustration and hyper-parameter tuning, multi-agent PG can obtain surprisingly robust efficiency in comparison with off-policy VD strategies.

Why might PG strategies work so nicely? On this publish, we are going to current concrete evaluation to indicate that in sure eventualities, e.g., environments with a extremely multi-modal reward panorama, VD may be problematic and result in undesired outcomes. In contrast, PG strategies with particular person insurance policies can converge to an optimum coverage in these circumstances. As well as, PG strategies with auto-regressive (AR) insurance policies can study multi-modal insurance policies.




Determine 1: completely different coverage illustration for the 4-player permutation sport.

CTDE in Cooperative MARL: VD and PG strategies

Centralized coaching and decentralized execution (CTDE) is a well-liked framework in cooperative MARL. It leverages international data for more practical coaching whereas retaining the illustration of particular person insurance policies for testing. CTDE may be carried out through worth decomposition (VD) or coverage gradient (PG), main to 2 several types of algorithms.

VD strategies study native Q networks and a mixing operate that mixes the native Q networks to a worldwide Q operate. The blending operate is normally enforced to fulfill the Particular person-World-Max (IGM) precept, which ensures the optimum joint motion may be computed by greedily selecting the optimum motion regionally for every agent.

In contrast, PG strategies immediately apply coverage gradient to study a person coverage and a centralized worth operate for every agent. The worth operate takes as its enter the worldwide state (e.g., MAPPO) or the concatenation of all of the native observations (e.g., MADDPG), for an correct international worth estimate.

The permutation sport: a easy counterexample the place VD fails

We begin our evaluation by contemplating a stateless cooperative sport, specifically the permutation sport. In an $N$-player permutation sport, every agent can output $N$ actions ${ 1,ldots, N }$. Brokers obtain $+1$ reward if their actions are mutually completely different, i.e., the joint motion is a permutation over $1, ldots, N$; in any other case, they obtain $0$ reward. Word that there are $N!$ symmetric optimum methods on this sport.




Determine 2: the 4-player permutation sport.

Allow us to concentrate on the 2-player permutation sport for our dialogue. On this setting, if we apply VD to the sport, the worldwide Q-value will factorize to

[Q_textrm{tot}(a^1,a^2)=f_textrm{mix}(Q_1(a^1),Q_2(a^2)),]

the place $Q_1$ and $Q_2$ are native Q-functions, $Q_textrm{tot}$ is the worldwide Q-function, and $f_textrm{combine}$ is the blending operate that, as required by VD strategies, satisfies the IGM precept.




Determine 3: high-level instinct on why VD fails within the 2-player permutation sport.

We formally show that VD can not symbolize the payoff of the 2-player permutation sport by contradiction. If VD strategies have been capable of symbolize the payoff, we’d have

[Q_textrm{tot}(1, 2)=Q_textrm{tot}(2,1)=1 qquad textrm{and} qquad Q_textrm{tot}(1, 1)=Q_textrm{tot}(2,2)=0.]

Nonetheless, if both of those two brokers have completely different native Q values, e.g. $Q_1(1)> Q_1(2)$, then in response to the IGM precept, we will need to have

[1=Q_textrm{tot}(1,2)=argmax_{a^2}Q_textrm{tot}(1,a^2)>argmax_{a^2}Q_textrm{tot}(2,a^2)=Q_textrm{tot}(2,1)=1.]

In any other case, if $Q_1(1)=Q_1(2)$ and $Q_2(1)=Q_2(2)$, then

[Q_textrm{tot}(1, 1)=Q_textrm{tot}(2,2)=Q_textrm{tot}(1, 2)=Q_textrm{tot}(2,1).]

Consequently, worth decomposition can not symbolize the payoff matrix of the 2-player permutation sport.

What about PG strategies? Particular person insurance policies can certainly symbolize an optimum coverage for the permutation sport. Furthermore, stochastic gradient descent can assure PG to converge to considered one of these optima below gentle assumptions. This means that, although PG strategies are much less widespread in MARL in contrast with VD strategies, they are often preferable in sure circumstances which can be widespread in real-world functions, e.g., video games with a number of technique modalities.

We additionally comment that within the permutation sport, with the intention to symbolize an optimum joint coverage, every agent should select distinct actions. Consequently, a profitable implementation of PG should be sure that the insurance policies are agent-specific. This may be completed through the use of both particular person insurance policies with unshared parameters (known as PG-Ind in our paper), or an agent-ID conditioned coverage (PG-ID).

Going past the straightforward illustrative instance of the permutation sport, we lengthen our examine to widespread and extra life like MARL benchmarks. Along with StarCraft Multi-Agent Problem (SMAC), the place the effectiveness of PG and agent-conditioned coverage enter has been verified, we present new ends in Google Analysis Soccer (GRF) and multi-player Hanabi Problem.





Determine 4: (left) successful charges of PG strategies on GRF; (proper) finest and common analysis scores on Hanabi-Full.

In GRF, PG strategies outperform the state-of-the-art VD baseline (CDS) in 5 eventualities. Curiously, we additionally discover that particular person insurance policies (PG-Ind) with out parameter sharing obtain comparable, typically even increased successful charges, in comparison with agent-specific insurance policies (PG-ID) in all 5 eventualities. We consider PG-ID within the full-scale Hanabi sport with various numbers of gamers (2-5 gamers) and examine them to SAD, a powerful off-policy Q-learning variant in Hanabi, and Worth Decomposition Networks (VDN). As demonstrated within the above desk, PG-ID is ready to produce outcomes akin to or higher than the perfect and common rewards achieved by SAD and VDN with various numbers of gamers utilizing the identical variety of setting steps.

Past increased rewards: studying multi-modal habits through auto-regressive coverage modeling

Moreover studying increased rewards, we additionally examine the right way to study multi-modal insurance policies in cooperative MARL. Let’s return to the permutation sport. Though we now have proved that PG can successfully study an optimum coverage, the technique mode that it lastly reaches can extremely rely upon the coverage initialization. Thus, a pure query can be:


Can we study a single coverage that may cowl all of the optimum modes?

Within the decentralized PG formulation, the factorized illustration of a joint coverage can solely symbolize one specific mode. Due to this fact, we suggest an enhanced option to parameterize the insurance policies for stronger expressiveness — the auto-regressive (AR) insurance policies.




Determine 5: comparability between particular person insurance policies (PG) and auto-regressive insurance policies (AR) within the 4-player permutation sport.

Formally, we factorize the joint coverage of $n$ brokers into the type of

[pi(mathbf{a} mid mathbf{o}) approx prod_{i=1}^n pi_{theta^{i}} left( a^{i}mid o^{i},a^{1},ldots,a^{i-1} right),]

the place the motion produced by agent $i$ relies upon by itself remark $o_i$ and all of the actions from earlier brokers $1,dots,i-1$. The auto-regressive factorization can symbolize any joint coverage in a centralized MDP. The solely modification to every agent’s coverage is the enter dimension, which is barely enlarged by together with earlier actions; and the output dimension of every agent’s coverage stays unchanged.

With such a minimal parameterization overhead, AR coverage considerably improves the illustration energy of PG strategies. We comment that PG with AR coverage (PG-AR) can concurrently symbolize all optimum coverage modes within the permutation sport.




Determine: the heatmaps of actions for insurance policies discovered by PG-Ind (left) and PG-AR (center), and the heatmap for rewards (proper); whereas PG-Ind solely converge to a particular mode within the 4-player permutation sport, PG-AR efficiently discovers all of the optimum modes.

In additional complicated environments, together with SMAC and GRF, PG-AR can study attention-grabbing emergent behaviors that require robust intra-agent coordination that will by no means be discovered by PG-Ind.





Determine 6: (left) emergent habits induced by PG-AR in SMAC and GRF. On the 2m_vs_1z map of SMAC, the marines preserve standing and assault alternately whereas making certain there is just one attacking marine at every timestep; (proper) within the academy_3_vs_1_with_keeper situation of GRF, brokers study a “Tiki-Taka” type habits: every participant retains passing the ball to their teammates.

Discussions and Takeaways

On this publish, we offer a concrete evaluation of VD and PG strategies in cooperative MARL. First, we reveal the limitation on the expressiveness of widespread VD strategies, exhibiting that they might not symbolize optimum insurance policies even in a easy permutation sport. In contrast, we present that PG strategies are provably extra expressive. We empirically confirm the expressiveness benefit of PG on widespread MARL testbeds, together with SMAC, GRF, and Hanabi Problem. We hope the insights from this work may benefit the group in direction of extra basic and extra highly effective cooperative MARL algorithms sooner or later.


This publish is predicated on our paper in joint with Zelai Xu: Revisiting Some Widespread Practices in Cooperative Multi-Agent Reinforcement Studying (paper, web site).

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