In cooperative multi-agent reinforcement studying (MARL), on account of 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 recent empirical studies show that with correct enter illustration and hyper-parameter tuning, multi-agent PG can obtain surprisingly strong performance in comparison with off-policy VD strategies.

**Why might PG strategies work so nicely?** On this submit, we are going to 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 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 simpler coaching whereas protecting the illustration of particular person insurance policies for testing. CTDE may be carried out 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 worldwide Q operate. The blending operate is often enforced to fulfill the Particular person-International-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 straight 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 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 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.

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

Allow us to concentrate on the 2-player permutation sport 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 characterize the payoff of the 2-player permutation sport by contradiction. If VD strategies had been capable of characterize the payoff, we’d 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’ve $argmax_{a^1}Q_1(a^1)=1$. Then in accordance with 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).]

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 among these optima under mild assumptions. This implies that, regardless that PG strategies are much less fashionable in MARL in contrast with VD strategies, they are often preferable in sure instances which are frequent in real-world purposes, e.g., video games with a number of technique modalities.

We additionally comment that within the permutation sport, with the intention 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 carried out 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 outperforms current VD strategies on fashionable MARL testbeds

Going past the easy illustrative instance of the permutation sport, we prolong our research to fashionable and extra practical 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 Challenge.

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. 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 eventualities. 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 surroundings steps.

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

Moreover studying larger rewards, we additionally research the way to 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 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 characterize one explicit mode. Due to this fact, 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 remark $o_i$ 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 discovered by PG-Ind (left) and PG-AR (center), and the heatmap for rewards (proper); whereas PG-Ind solely converge to a particular 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 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 hold 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 » fashion 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 fashionable VD strategies, exhibiting that they may not characterize 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 fashionable 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 is predicated on our paper: Revisiting Some Widespread Practices in Cooperative Multi-Agent Reinforcement Studying (paper, website).*