In cooperative multi-agent reinforcement studying (MARL), as a consequence of its *on-policy* nature, coverage gradient (PG) strategies are sometimes believed to be much less pattern environment friendly than worth decomposition (VD) strategies, that are *off-policy*. Nevertheless, some latest empirical research exhibit 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’ll current concrete evaluation to point out that in sure situations, e.g., environments with a extremely multi-modal reward panorama, VD could 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 be taught 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 holding the illustration of particular person insurance policies for testing. CTDE could be carried out 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 world Q perform. The blending perform is often enforced to fulfill the Particular person-World-Max (IGM) precept, which ensures the optimum joint motion could be computed by greedily selecting the optimum motion regionally for every agent.

Against this, PG strategies immediately 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 sport: a easy counterexample the place VD fails

We begin our evaluation by contemplating a stateless cooperative sport, particularly 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 deal with 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 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 sport.

We formally show that VD can’t signify the payoff of the 2-player permutation sport by contradiction. If VD strategies have been capable of signify 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.]

Nevertheless, 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).]

In consequence, worth decomposition can’t signify the payoff matrix of the 2-player permutation sport.

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

We additionally comment that within the permutation sport, in an effort to signify an optimum joint coverage, every agent should select distinct actions. **Consequently, a profitable implementation of PG should be certain that the insurance policies are agent-specific.** This may be achieved 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 outperform greatest VD strategies on common MARL testbeds

Going past the straightforward illustrative instance of the permutation sport, we lengthen our examine to common 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 situations. Curiously, we additionally discover that particular person insurance policies (PG-Ind) with out parameter sharing obtain comparable, generally even larger profitable charges, in comparison with agent-specific insurance policies (PG-ID) in all 5 situations. We consider PG-ID within the full-scale Hanabi sport 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 one of the best and common rewards achieved by SAD and VDN with various numbers of gamers utilizing the identical variety of setting steps.

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

Moreover studying larger rewards, we additionally examine the best way to be taught multi-modal insurance policies in cooperative MARL. Let’s return to the permutation sport. Though we’ve got 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 will probably 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 signify one explicit mode. Subsequently, we suggest an enhanced strategy 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 commentary $o_i$ and all of the actions from earlier brokers $1,dots,i-1$. The auto-regressive factorization can signify *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 signify all optimum coverage modes within the permutation sport.

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 sport, 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 robust 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 making certain 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” fashion 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 common VD strategies, exhibiting that they might not signify optimum insurance policies even in a easy permutation sport. Against this, we present that PG strategies are provably extra expressive. We empirically confirm the expressiveness benefit of PG on common MARL testbeds, together with SMAC, GRF, and Hanabi Problem. We hope the insights from this work may benefit the group in direction of extra normal and extra highly effective cooperative MARL algorithms sooner or later.

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