In cooperative multi-agent reinforcement studying (MARL), attributable to 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 show 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 may PG strategies work so nicely?** On this publish, we’ll 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. Against this, PG strategies with particular person insurance policies can converge to an optimum coverage in these instances. 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 recreation.

## 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* info for simpler coaching whereas preserving the illustration of particular person insurance policies for testing. CTDE may be applied by way of 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 world 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.

Against this, 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 recreation: a easy counterexample the place VD fails

We begin our evaluation by contemplating a stateless cooperative recreation, specifically the permutation recreation. In an $N$-player permutation recreation, 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. Be aware that there are $N!$ symmetric optimum methods on this recreation.

Determine 2: the 4-player permutation recreation.

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

Allow us to concentrate on the 2-player permutation recreation now and apply VD to the sport. On this stateless setting, we use $Q_1$ and $Q_2$ to indicate the native Q-functions, and use $Q_textrm{tot}$ to indicate the worldwide Q-function. The IGM precept requires that

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

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

[Q_textrm{tot}(1, 2)=Q_textrm{tot}(2,1)=1quad text{and}quad Q_textrm{tot}(1, 1)=Q_textrm{tot}(2,2)=0.]

If both of those two brokers has completely different native Q values (e.g. $Q_1(1)> Q_1(2)$), we now have $argmax_{a^1}Q_1(a^1)=1$. Then in response to the IGM precept, *any* optimum joint motion

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

satisfies $a^{1star}=1$ and $a^{1star}neq 2$, so the joint motion $(a^1,a^2)=(2,1)$ is sub-optimal, i.e., $Q_textrm{tot}(2,1)<1$.

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).]

In consequence, worth decomposition can not symbolize the payoff matrix of the 2-player permutation recreation.

What about PG strategies? Particular person insurance policies can certainly symbolize an optimum coverage for the permutation recreation. Furthermore, stochastic gradient descent can assure PG to converge to one in every of these optima beneath gentle assumptions. This implies that, though PG strategies are much less well-liked in MARL in contrast with VD strategies, they are often preferable in sure instances 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 recreation, with a view to symbolize an optimum joint coverage, every agent should select distinct actions. **Consequently, a profitable implementation of PG should make sure that the insurance policies are agent-specific.** This may be completed by utilizing both particular person insurance policies with unshared parameters (known as PG-Ind in our paper), or an agent-ID conditioned coverage (PG-ID).

## PG outperforms present VD strategies on well-liked MARL testbeds

Going past the easy illustrative instance of the permutation recreation, we prolong our examine to well-liked and extra real looking 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) greatest and common analysis scores on Hanabi-Full.

In GRF, PG strategies outperform the state-of-the-art VD baseline (CDS) in 5 eventualities. Apparently, we additionally discover that particular person insurance policies (PG-Ind) with out parameter sharing obtain comparable, typically even greater successful charges, in comparison with agent-specific insurance policies (PG-ID) in all 5 eventualities. We consider PG-ID within the full-scale Hanabi recreation with various numbers of gamers (2-5 gamers) and evaluate 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 similar to or higher than the perfect and common rewards achieved by SAD and VDN with various numbers of gamers utilizing the identical variety of surroundings steps.

## Past greater rewards: studying multi-modal conduct by way of auto-regressive coverage modeling

Moreover studying greater rewards, we additionally examine find out how to study multi-modal insurance policies in cooperative MARL. Letâ€™s return to the permutation recreation. Though we now have proved that PG can successfully study an optimum coverage, the technique mode that it lastly reaches can extremely depend upon the coverage initialization. Thus, a pure query will likely 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 explicit mode. Subsequently, we suggest an enhanced solution to parameterize the insurance policies for stronger expressiveness â€” 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 recreation.

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 commentary $o_i$ 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 agentâ€™s coverage is the enter dimension, which is barely enlarged by together with earlier actions; and the output dimension of every agentâ€™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 recreation.

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 selected mode within the 4-player permutation recreation, PG-AR efficiently discovers all of the optimum modes.

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

Determine 6: (left) emergent conduct induced by PG-AR in SMAC and GRF. On the 2m_vs_1z map of SMAC, the marines maintain standing and assault alternately whereas guaranteeing there is just one attacking marine at every timestep; (proper) within the academy_3_vs_1_with_keeper state of affairs of GRF, brokers study a “Tiki-Taka” type conduct: 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 well-liked VD strategies, exhibiting that they may not symbolize optimum insurance policies even in a easy permutation recreation. Against this, we present that PG strategies are provably extra expressive. We empirically confirm the expressiveness benefit of PG on well-liked MARL testbeds, together with SMAC, GRF, and Hanabi Problem. We hope the insights from this work may benefit the neighborhood in the direction of extra common and extra highly effective cooperative MARL algorithms sooner or later.

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