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 exhibit 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 might 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 could 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 study multi-modal insurance policies.

Determine 1: totally 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 *world* info for more practical coaching whereas conserving the illustration of particular person insurance policies for testing. CTDE could 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 worldwide Q perform. The blending perform is normally 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 domestically for every agent.

In contrast, 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 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 totally 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 sport.

Determine 2: the 4-player permutation sport.

Allow us to give attention to 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 not characterize the payoff of the 2-player permutation sport by contradiction. If VD strategies had been in a position to characterize 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 totally different native Q values, e.g. $Q_1(1)> Q_1(2)$, then based on 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).]

Because of this, worth decomposition can not characterize the payoff matrix of the 2-player permutation sport.

What about PG strategies? Particular person insurance policies can certainly characterize an optimum coverage for the permutation sport. Furthermore, stochastic gradient descent can assure PG to converge to one in every of these optima below 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 functions, e.g., video games with a number of technique modalities.

We additionally comment that within the permutation sport, with a view to characterize 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 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 finest VD strategies on standard MARL testbeds

Going past the straightforward illustrative instance of the permutation sport, we lengthen our examine to standard 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 leads to 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 situations. Curiously, we additionally discover that particular person insurance policies (PG-Ind) with out parameter sharing obtain comparable, typically even larger successful 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 corresponding 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 habits through auto-regressive coverage modeling

In addition to studying larger rewards, we additionally examine study multi-modal insurance policies in cooperative MARL. Let’s return to the permutation sport. Though we’ve 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 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 characterize one explicit mode. Subsequently, we suggest an enhanced solution 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 characterize *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 characterize 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 study fascinating emergent behaviors that require sturdy intra-agent coordination which will by no means be realized 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 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 study a “Tiki-Taka” fashion habits: 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 standard VD strategies, displaying that they might not characterize 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 standard MARL testbeds, together with SMAC, GRF, and Hanabi Problem. We hope the insights from this work may gain advantage the group in the direction of extra normal 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).*