Why do Coverage Gradient Strategies work so properly in Cooperative MARL? Proof from Coverage Illustration

In cooperative multi-agent reinforcement studying (MARL), as a result 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 display 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 properly? On this submit, we are going to 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 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 info for more practical coaching whereas conserving the illustration of particular person insurance policies for testing. CTDE will be applied through worth decomposition (VD) or coverage gradient (PG), main to 2 various kinds of algorithms.

VD strategies be taught native Q networks and a mixing operate that mixes the native Q networks to a world Q operate. The blending operate is normally enforced to fulfill the Particular person-World-Max (IGM) precept, which ensures the optimum joint motion will be computed by greedily selecting the optimum motion domestically for every agent.

Against this, PG strategies instantly apply coverage gradient to be taught 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 international 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. Be aware 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 symbolize the payoff of the 2-player permutation sport by contradiction. If VD strategies have been in a position to symbolize 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. $Q_1(1)> Q_1(2)$), now we have $argmax_{a^1}Q_1(a^1)=1$. 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).]

Consequently, worth decomposition can not symbolize the payoff matrix of the 2-player permutation sport.

What about PG strategies? Particular person insurance policies can certainly symbolize an optimum coverage for the permutation sport. Furthermore, stochastic gradient descent can assure PG to converge to one in all these optima below delicate 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 might be widespread 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 symbolize 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 finished 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 straightforward illustrative instance of the permutation sport, we prolong 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. Curiously, 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 sport with various numbers of gamers (2-5 gamers) and evaluate 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 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 atmosphere steps.

Apart from studying larger rewards, we additionally examine find out how to be taught multi-modal insurance policies in cooperative MARL. Let’s return to the permutation sport. Though now we have proved that PG can successfully be taught 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 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 symbolize one explicit mode. Due to this fact, 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 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 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 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 complicated 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 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 preserve 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 state of affairs of GRF, brokers be taught 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 fashionable VD strategies, exhibiting that they might not symbolize 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 direction of extra basic and extra highly effective cooperative MARL algorithms sooner or later.

This submit relies on our paper: Revisiting Some Frequent Practices in Cooperative Multi-Agent Reinforcement Studying (paper, web site).