In cooperative multi-agent reinforcement studying (MARL), because 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 current 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 nicely?** On this publish, we are going to current concrete evaluation to indicate 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 circumstances. 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 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* info for simpler coaching whereas protecting 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 various kinds 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 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 instantly 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 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 totally 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.

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

Allow us to give attention to the 2-player permutation recreation 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’t symbolize the payoff of the 2-player permutation recreation by contradiction. If VD strategies have been capable of symbolize the payoff, we might 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 totally different native Q values (e.g. $Q_1(1)> Q_1(2)$), we’ve got $argmax_{a^1}Q_1(a^1)=1$. Then in response to 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).]

In consequence, 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 among these optima underneath delicate assumptions. This implies that, though PG strategies are much less fashionable 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, so as to symbolize an optimum joint coverage, every agent should select distinct actions. **Consequently, a profitable implementation of PG should make sure that the insurance policies are agent-specific.** This may be achieved 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 present VD strategies on fashionable MARL testbeds

Going past the straightforward illustrative instance of the permutation recreation, we lengthen our examine 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 leads to Google Analysis Soccer (GRF) and multi-player Hanabi Problem.

Determine 4: (left) successful 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. Apparently, we additionally discover that particular person insurance policies (PG-Ind) with out parameter sharing obtain comparable, generally 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 recreation with various numbers of gamers (2-5 gamers) and examine 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 setting steps.

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

Moreover studying larger rewards, we additionally examine learn how to study multi-modal insurance policies in cooperative MARL. Let’s return to the permutation recreation. Though we’ve got proved that PG can successfully study an optimum coverage, the technique mode that it lastly reaches can extremely rely upon 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 symbolize one explicit mode. Due to this fact, we suggest an enhanced method 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 statement $o_i$ 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 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 symbolize 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 particular 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 sturdy intra-agent coordination that will by no means be discovered 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 preserve 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” model habits: 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 fashionable VD strategies, exhibiting that they may 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 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 common and extra highly effective cooperative MARL algorithms sooner or later.

*This publish is predicated on our paper: Revisiting Some Widespread Practices in Cooperative Multi-Agent Reinforcement Studying (paper, web site).*