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*. Nevertheless, some latest empirical research display that with correct enter illustration and hyper-parameter tuning, multi-agent PG can obtain surprisingly sturdy efficiency in comparison with off-policy VD strategies.

**Why may PG strategies work so properly?** On this put up, we are going to current concrete evaluation to indicate that in sure eventualities, e.g., environments with a extremely multi-modal reward panorama, VD will be problematic and result in undesired outcomes. In contrast, 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 be taught 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* data for more practical coaching whereas maintaining the illustration of particular person insurance policies for testing. CTDE will be applied by way of worth decomposition (VD) or coverage gradient (PG), main to 2 several types of algorithms.

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

In contrast, PG strategies straight apply coverage gradient to be taught a person coverage and a centralized worth perform for every agent. The worth perform 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, particularly 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. Word that there are $N!$ symmetric optimum methods on this recreation.

Determine 2: the 4-player permutation recreation.

Allow us to deal with the 2-player permutation recreation 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 perform that, as required by VD strategies, satisfies the IGM precept.

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

We formally show that VD can’t symbolize the payoff of the 2-player permutation recreation by contradiction. If VD strategies had been capable of symbolize the payoff, we might 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.]

Nevertheless, if both of those two brokers have completely different native Q values, e.g. $Q_1(1)> Q_1(2)$, then in keeping with the IGM precept, we should 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).]

Because of this, worth decomposition can’t 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 all these optima beneath gentle assumptions. This means that, regardless that PG strategies are much less standard in MARL in contrast with VD strategies, they are often preferable in sure instances which are widespread in real-world purposes, e.g., video games with a number of technique modalities.

We additionally comment that within the permutation recreation, to be able 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 accomplished 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).

## PG outperform greatest VD strategies on standard MARL testbeds

Going past the straightforward illustrative instance of the permutation recreation, we lengthen our examine to standard 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) profitable 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 larger profitable 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 robust 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 larger rewards: studying multi-modal conduct by way of auto-regressive coverage modeling

Apart from studying larger rewards, we additionally examine easy methods to be taught multi-modal insurance policies in cooperative MARL. Letâ€™s return to the permutation recreation. Though now we have proved that PG can successfully be taught an optimum coverage, the technique mode that it lastly reaches can extremely depend upon the coverage initialization. Thus, a pure query can be:

Can we be taught 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 strategy 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 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 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 realized 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 be taught attention-grabbing emergent behaviors that require sturdy intra-agent coordination which will by no means be realized 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 situation of GRF, brokers be taught a “Tiki-Taka” type conduct: every participant retains passing the ball to their teammates.

## Discussions and Takeaways

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

*This put up relies on our paper in joint with Zelai Xu: Revisiting Some Widespread Practices in Cooperative Multi-Agent Reinforcement Studying (paper, web site).*