In cooperative multi-agent reinforcement studying (MARL), on account 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*. Nonetheless, some latest empirical research reveal 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 effectively?** On this submit, we’ll current concrete evaluation to point out that in sure situations, e.g., environments with a extremely multi-modal reward panorama, VD will be problematic and result in undesired outcomes. Against this, 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 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 *world* data for simpler coaching whereas retaining the illustration of particular person insurance policies for testing. CTDE will 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 perform that mixes the native Q networks to a world 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 regionally for every agent.

Against this, PG strategies straight apply coverage gradient to study 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 world 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. Notice 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 signify the payoff of the 2-player permutation recreation by contradiction. If VD strategies had been in a position to signify 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.]

Nonetheless, if both of those two brokers have completely different native Q values, e.g. $Q_1(1)> Q_1(2)$, then based on 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).]

Consequently, worth decomposition can’t signify the payoff matrix of the 2-player permutation recreation.

What about PG strategies? Particular person insurance policies can certainly signify an optimum coverage for the permutation recreation. Furthermore, stochastic gradient descent can assure PG to converge to one in all these optima underneath delicate assumptions. This means that, regardless that PG strategies are much less in style in MARL in contrast with VD strategies, they are often preferable in sure circumstances 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 recreation, to be able 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 finished 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 finest VD strategies on in style MARL testbeds

Going past the straightforward illustrative instance of the permutation recreation, we prolong our examine to in style and extra reasonable 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) finest and common analysis scores on Hanabi-Full.

In GRF, PG strategies outperform the state-of-the-art VD baseline (CDS) in 5 situations. 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 situations. We consider PG-ID within the full-scale Hanabi recreation with various numbers of gamers (2-5 gamers) and examine 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 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 larger rewards: studying multi-modal conduct through auto-regressive coverage modeling

Moreover studying larger rewards, we additionally examine the best way to study multi-modal insurance policies in cooperative MARL. Let’s return to the permutation recreation. Though now we have proved that PG can successfully study an optimum coverage, the technique mode that it lastly reaches can extremely rely on the coverage initialization. Thus, a pure query shall 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 signify one specific mode. Subsequently, 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 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 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 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 complicated environments, together with SMAC and GRF, PG-AR can study fascinating emergent behaviors that require robust intra-agent coordination that 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 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 conduct: every participant retains passing the ball to their teammates.

## Discussions and Takeaways

On this submit, we offer a concrete evaluation of VD and PG strategies in cooperative MARL. First, we reveal the limitation on the expressiveness of in style VD strategies, displaying that they may not signify 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 in style MARL testbeds, together with SMAC, GRF, and Hanabi Problem. We hope the insights from this work may gain advantage the neighborhood in the direction of extra common and extra highly effective cooperative MARL algorithms sooner or later.

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