Permutation Invariant Policy Optimization for Mean-Field Multi-Agent Reinforcement Learning: A Principled Approach

Multi-Agent Reinforcement Learning (Littman, 1994; Zhang et al., 2019) generalizes Reinforcement Learning (Sutton and Barto, 2018) to address the sequential decision-making problem of multiple agents maximizing their own long term rewards while interacting with one another in a common environment. With breakthroughs in deep learning, MARL algorithms equipped with deep neural networks have seen significant empirical successes in various domains, including simulated autonomous driving (Shalev-Shwartz et al., 2016), multi-agent robotic control (Matarić, 1997; Kober et al., 2013), and E-sports (Vinyals et al., 2019).

Despite tremendous successes, MARL is notoriously hard to scale to the many-agent setting, as the size of the state-action space grows exponentially with respect to the number of agents. This phenomenon is recently described as the curse of many agents (Menda et al., 2018). To tackle this challenge, we focus on cooperative MARL, where agents work together to maximize their team reward (Panait and Luke, 2005). We identify and exploit a key property of cooperative MARL with homogeneous agents, namely the invariance with respect to the permutation of agents. Such permutation invariance can be found in many real-world scenarios with homogeneous agents, such as distributed control of multiple autonomous vehicles and team sports (Cao et al., 2013; Kalyanakrishnan et al., 2006), and also in the case of heterogeneous groups where invariance holds within each group (Liu et al., 2019b). More importantly, we further find that permutation invariance has significant practical implications, as the optimal value functions remains invariant when permuting the joint state-action pairs. Such an observation strongly advocates a permutation invariant design for learning, which helps reduce the effective search space of the policy/value functions from exponential dependence on the number of agents to polynomial dependence.

Several empirical methods have been proposed to incorporate permutation invariance into solving MARL problems. Liu et al. (2019b) implement a permutation invariant critic based on Graph Convolutional Network (GCN) (Kipf and Welling, 2017). Sunehag et al. (2017) propose value decomposition, which together with parameter sharing, leads to a joint critic network that is permutation invariant over agents. While these methods are based on heuristics, we are the first to provide theoretical principles for introducing permutation invariance as an inductive bias for learning value functions and policy in homogeneous systems. In addition, we adopt the DeepSet (Zaheer et al., 2017) architecture, which is well suited for handling homogeneity of agents, with much simpler operations to induce permutation invariance, and being much more parameter efficient by doing so.

To scale up learning in MARL problem in the presence of a large number, even infinitely many agents, mean-field approximation has been explored to directly model the population behavior of the agents. Yang et al. (2017) consider a mean-field game with deterministic linear state transitions, and show it can be reformulated as a mean-field MDP, with mean-field state residing in a finite-dimensional probability simplex. Yang et al. (2018) take the mean-field approximation over actions, such that the interaction for any given agent and the population is approximated by the interaction between the agent’s action and the averaged actions of its neighboring agents. However, the motivation for averaging over local actions remains unclear, and it generally requires a sparse graph over agents. In practice, properly identifying such structure also demands extensive prior knowledge. Carmona et al. (2019) motivate a mean-field MDP from the perspective of mean-field control. The mean-field state therein lies in a probability simplex and thus continuous in nature. To enable the ensuing Q-learning algorithm, discretization of the joint state-action space is necessary. Wang et al. (2020) motivate a mean-field MDP from permutation invariance, but assume a central controller coordinating the actions of all the agents, and hence is restricted to handling the curse of many agents from the exponential blowup of joint state space. In contrast, our formulation of mean-field approximation allows agents to make their own local actions without resorting to a centralized controller.

We propose a mean-field Markov decision process motivated from the permutation invariance structure of cooperative MARL, which can be viewed as a natural limit of finite-agent MDP by taking the number of agents to infinity. Such a mean-field MDP generalizes traditional MDP, with each state representing a distribution over the state space of a single agent. The mean-field MDP provides us a tractable formulation to model MDP with many agents, even infinite number of agents. We further propose the Mean-Field Proximal Policy Optimization (MF-PPO) algorithm, at the core of which is a pair of permutation invariant actor and critic neural networks. These networks are implemented based on DeepSet (Zaheer et al., 2017), which uses convolutional type operations to induce permutation invariance over the set of inputs. We show that with sufficiently many agents, MF-PPO converges to the optimal policy of the mean-field MDP with a sublinear sample complexity independent of the number of agents. To support our theory, we conduct numerical experiments on the benchmark multi-agent particle environment (MPE) and show that our proposed method requires a smaller number of model parameters and attains a better performance compared to multiple baselines.

Notations. For a set $X$, we denote $\mathcal{P}(X)$ as the set of distribution on $X$. $\delta_{x}$ denotes the Dirac measure supported at $x$. For $\mathbf{s}=(s_{1},\ldots,s_{N})$, we use $\mathbf{s}\overset{\mathrm{i.i.d.}}{\sim}p$ to denote that each $s_{i}$ is independently sampled from distribution $p$. For a function $f:X\to\mathbb{R}$ and a distribution $\pi\in\mathcal{P}(X)$, we write $\left\langle f,\pi\right\rangle=\mathbb{E}_{a\sim\pi}f(a)$. We write $[m]$ in short for $\{1,\ldots,m\}$.

We focus on studying multi-agent systems with cooperative, homogeneous agents, where the agents within the system are of similar nature and hence cannot be distinguished from each other. Specifically, we consider a discrete time control problem with $N$ agents, formulated as a Markov decision process $({\mathcal{S}}^{N},\mathcal{A}^{N},\mathbb{P},\mathrm{r})$. We define the joint state space ${\mathcal{S}}^{N}$ to be the Cartesian product of the finite state space ${\mathcal{S}}$ for each agent, and similarly define joint action space $\mathcal{A}^{N}$. The homogeneous nature of the system is reflected in the transition kernel $\mathbb{P}$ and the shared reward $\mathrm{r}$, which satisfies:

for all $(\mathbf{s}_{t},\mathbf{a}_{t})\in{\mathcal{S}}^{N}\times\mathcal{A}^{N}$ and permutation mapping $\kappa(\cdot)$. In other words, it is the configuration, rather than the identities of the agents, that affects the team reward, and the transition to the next configuration solely depends on the current configuration. See Figure 1 for a detailed illustration. Such permutation invariance can be found in many real-world scenarios, such as distributed control of multiple autonomous vehicles, team sports and social economic systems (Zheng et al., 2020; Cao et al., 2013; Kalyanakrishnan et al., 2006).

Our goal is to find the optimal policy $\nu$ within a policy class which maximizes the expected discounted reward

Our first result shows that learning with permutation invariance advocates permutation invariant network design.

Proposition 2.1 has an important implication for architecture design, as it states that it suffices to search within the permutation invariant policy class and value function class. To the best of our knowledge, this is the first theoretical justification of permutation invariant network design for learning with homogeneous agents.

We focus on the factorized policy class with parameter sharing scheme, where each agent makes its own decision given local state. Specifically, the joint policy $\nu$ can be factorized as $\nu(\mathbf{a}|\mathbf{s})=\prod_{i=1}^{N}\mu(a_{i}|s_{i})$, where $\mu(\cdot)$ denotes the shared local mapping. Such a policy class is widely adopted in the celebrated centralized training – decentralized execution paradigm (Lowe et al., 2017), due to its light overhead in the deployment phase and favorable performances. However, directly learning such factorized policy remains challenging, as each agent needs to estimate its state-action value function, denoted as $Q^{\nu}(\mathbf{s},\mathbf{a})$. The search space during learning is $(|{\mathcal{S}}|\times|\mathcal{A}|)^{N}$, which scales exponentially with respect to the number of agents. The large search space poses as a significant roadblock for efficient learning, and was recently coined as the curse of many agents (Menda et al., 2018).

To address the curse of many agents, we exploit the homogeneity of the system and take mean-field approximation. We begin by taking the perspective of agent $i$, which is arbitrarily chosen from the $N$ agents. We denote the state of such agent by $s$ and the states of the rest of the agents by $\mathbf{s}_{r}$. One can verify that when permuting the state of all the other agents, the value function remains unchanged; additionally, we can further characterize the value function as a function of the local state, and the empirical state distribution over the rest of agents.

For a system with a large number of agents (e.g., financial markets, social networks), the empirical state distribution can be seen as the concrete realization of the underlying distribution of the population. Motivated from this observation and Proposition 2.2, we formulate the following mean-field MDP that can be seen as the limit of finite-agent MDP in the presence of infinitely many homogeneous agents.

The defined mean-field MDP has an intimate connection with our previously discussed finite-agent MDP: here $(s,\mathrm{d}_{\mathcal{S}})$ represents the joint state viewed from the individual agent with state $s$; each agent makes a local decision using the shared local mapping $\overline{a}:{\mathcal{S}}\to\mathcal{A}$ and receive a local reward, whose cumulative effect is captured by the team reward $\mathrm{r}(s,\mathrm{d}_{\mathcal{S}},\overline{a})$; finally, the joint state transits to $(s^{\prime},\mathrm{d}_{\mathcal{S}}^{\prime})$ according to kernel $\mathbb{P}(s^{\prime},\mathrm{d}_{\mathcal{S}}^{\prime}|s,\mathrm{d}_{\mathcal{S}},\overline{a})$. We can clearly see that the mean-field MDP has a step-to-step correspondence with the finite-agent MDP with homogeneous agents, thus making it a natural generalization in the presence of large number of agents.

Our goal is to learn efficiently a policy $\pi$ whose expected discounted reward is maximized. To facilitate ensuing discussions, we define the value function as

where $(s_{0},\mathrm{d}_{{\mathcal{S}},0})=(s,\mathrm{d}_{\mathcal{S}}),\overline{a}_{t}\sim\pi(s_{t},\mathrm{d}_{{\mathcal{S}},t}),\forall t\geq 0$, and Q-function as

where $(s_{0},\mathrm{d}_{{\mathcal{S}},0})=(s,\mathrm{d}_{\mathcal{S}}),\overline{a}_{0}=\overline{a},\overline{a}_{t}\sim\pi(s_{t},\mathrm{d}_{{\mathcal{S}},t})$. The optimal policy is then denoted by

Despite the intuitive analogy to finite-agent MDP, solving the mean-field MDP posses some unique challenges. In addition to the unknown transition kernel and reward, the mean-field MDP takes a distribution as its state, which we do not have complete information of during training. In the following section, we propose our mean-field Proximal Policy Optimization (MF-PPO) algorithm that, with a careful architecture design, can solve such mean-field MDP in a model-free fashion efficiently.

Our algorithm falls into the category of the actor-critic learning paradigm, consisting of alternating iterations of policy evaluation and improvement. The unique features of MF-PPO lie in the facts: (1) it exploits permutation invariance of the system, reducing search space of the actor/critic networks drastically and enables much more efficient learning; (2) it can handle a varying number of agents. For simplicity of exposition, we consider a fixed number of agents here.

Throughout the rest of the section, we assume that for any joint state $(s,\mathrm{d}_{{\mathcal{S}}})\in{\mathcal{S}}\times\mathcal{P}({\mathcal{S}})$, the agent has access to $N$ $\mathrm{i.i.d.}$ samples $\{s_{i}\}_{i=1}^{N}$ from $\mathrm{d}_{{\mathcal{S}}}$. We denote concatenation of such samples as $\mathbf{s}\in{\mathcal{S}}^{N}$ and write $\mathbf{s}\overset{\mathrm{i.i.d.}}{\sim}\mathrm{d}_{{\mathcal{S}}}$.

MF-PPO maintains a pair of actor (denoted by $F^{A}$) and critic networks (denoted by $F^{Q}$), and uses the actor network to induce an energy-based policy $\pi(\overline{a}|s,\mathrm{d}_{{\mathcal{S}}})$. Specifically, given state $(s,\mathrm{d}_{{\mathcal{S}}})$, the actor network induces a distribution on the action set $\overline{\mathcal{A}}$ according to

where $\tau$ denotes the temperature parameter. We use $\pi\propto\exp\left\{F^{A}\right\}$ to denote the dependency of the policy on the energy function.

Compared to architectures without permutation invariance, whose search space depends on $N$ at the order of $(|{\mathcal{S}}||\mathcal{A}|)^{N}$, we can clearly see the search space of MF-PPO can be exponentially smaller.

Motivated by the characterization of the permutation invariant set function in Zaheer et al. (2017), the actor/critic network in MF-PPO takes the form of Deep Sets architecture, i.e.,

Both networks first aggregate local information by averaging over the output of a shared sub-network among agents, before feeding the aggregated information into a subsequent network. See Figure 2 for a detailed illustration. Effectively, by the average pooling layer and the preceding parameter sharing scheme, the network can keep its output unchanged when permuting the ordering of agents. Compared to a Graph Convolutional Neural Network (Kipf and Welling, 2017), the averaging operations are well suited for homogeneous agents and more parameter-efficient.

Naturally, the actor network when given the joint state-action pair $(s,\mathrm{d}_{{\mathcal{S}}},\overline{a})$ is given by

We assume $F^{A}$ is parameterized by a neural network with parameters $\alpha\in\mathbb{R}^{D}$, which is to be learned during training. We let the function class of all possible actor networks be denoted by $\mathcal{F}^{A}$. This same architecture design applies to the critic network, with learnable parameters denoted by $\theta\in\mathbb{R}^{D}$ and the function class denoted by $\mathcal{F}^{Q}$. MF-PPO then consists of successive iterations of policy evaluation and policy improvement steps described in detail below.

$\bullet$ Policy Evaluation. At the $k$-th iteration of MF-PPO, we first update the critic network $F^{Q}$ by minimizing the mean squared Bellman error while holding the actor network $F^{A,k}$ fixed. We denote the policy induced by the actor network as $\pi_{k}$, the stationary state distribution of policy $\pi_{k}$ as $\nu_{k}$, and the stationary state-action distribution as $\sigma_{k}=\nu_{k}\pi_{k}$. Thus, we follow the update

where $\mathbb{B}(\theta_{0},R_{\theta})$ denotes the Euclidean ball with radius $R_{\theta}$ centered at the initialized parameter $\theta_{0}$, and the Bellman evaluation operator ${\mathcal{T}}^{\pi}$ is given by

where $\mathrm{d}^{\prime}_{{\mathcal{S}}}\sim\mathbb{P}(\cdot|\mathrm{d}_{{\mathcal{S}}},\overline{a}),\overline{a}^{\prime}\sim\pi(\cdot|s^{\prime},\mathrm{d}^{\prime}_{{\mathcal{S}}})$. We solve (2) by T-step temporal-difference (TD) update and output $\theta_{k}=\theta(T)$. At the $t$-th iteration of the TD update,

where we sample

We use $\Pi_{\mathrm{X}}(\cdot)$ to denote the orthogonal projection onto set $\mathrm{X}$, and $\eta$ to denote the step size. The details are summarized in Algorithm 2 in Appendix A.

$\bullet$ Policy Improvement. Following the policy evaluation step, MF-PPO updates its policy by updating the policy network $F^{A}$, which is the energy function associated with the policy. We update the policy network by

The update rule intuitively reads as increasing the probability for choosing action $\overline{a}$ if it yields a higher value for critic network $F^{Q}(s,\mathrm{d}_{{\mathcal{S}}},\overline{a})$, which can be viewed as a softened version of policy iteration (Bertsekas, 2011). Additionally, by controlling the proximity parameter $\upsilon_{k}$, we can control the softness of the update, with $\upsilon\to 0$ yielding the vanilla policy iteration. Moreover, without constraint of $F^{A,k+1}\in\mathcal{F}^{A}$, such an update would have a nice closed form expression, and $\pi_{k+1}$ itself is another energy-based policy.

To take into account that the representable function of actor network resides in $\mathcal{F}_{A}$, we update the policy by projecting the energy function of $\overline{\pi}_{k+1}$ back to $\mathcal{F}_{A}$. Specifically, by denoting

we recover the next actor network $F^{A}_{\alpha_{k+1}}$ (i.e., energy) by performing the following regression task

where $\widetilde{\sigma}_{k}=\nu_{k}\pi_{0}$. We approximately solve (3) via T-step stochastic gradient descent (SGD), and output

At the $t$-th iteration of SGD,

where we sample $(s,\mathrm{d}_{{\mathcal{S}}},\overline{a})\sim\widetilde{\sigma}_{k}$, and $\mathbf{s}\overset{\mathrm{i.i.d.}}{\sim}\mathrm{d}_{{\mathcal{S}}}$, and $\eta$ is the step size. The details are summarized in Algorithm 3 of Appendix A. Finally, we present the complete MF-PPO in Algorithm 1.

We present the global convergence of MF-PPO algorithm for two-layer permutation-invariant parameterization of actor and critic network. We remark that our analysis can be extended to multi-layer permutation-invariant networks, and we present the two-layer case here for simplicity of exposition. Specifically, the actor and critic networks take the form

where $m_{A}$ (resp. $m_{Q}$) is the width of the actor (resp. critic) network, and $\sigma(x)=\max\{x,0\}$ denotes the ReLU activation. We randomly initialize $u_{j}$ (resp. $v_{j}$) and first layer weights $\alpha_{0}=[\alpha_{0,1}^{\top},\ldots,\alpha_{0,m_{A}}^{\top}]^{\top}\in\mathbb{R}^{d\cdot m_{A}}$ (resp. $\theta_{0}\in\mathbb{R}^{d\cdot m_{Q}}$) by

For the ease of analysis, we take $m=m_{A}=m_{Q}$ and share the initialization of $\alpha_{0}$ and $\theta_{0}$ (resp. $u_{0}$ and $v_{0}$). Additionally, we keep $u_{j}$’s fixed during training, and $\alpha$ (resp. $\theta$) within the ball $\mathbb{B}(\alpha_{0},R_{A})$ (resp. $\mathbb{B}(\theta_{0},R_{Q})$) throughout training. We define the following function class characterizing the class of previously defined actor/critic network for large network width.

It is well known that functions within $\mathcal{F}_{\beta,m}$ approximate functions within the reproducing kernel Hilbert space associated with kernel $\mathrm{K}(x,y)=\mathbb{E}_{z\sim\mathcal{N}(0,I_{d}/d)}\left\{\mathbbm{1}(z^{\top}x>0,z^{\top}y>0)\right\}$ for large network width $m$ (Jacot et al., 2018; Chizat and Bach, 2018; Cai et al., 2019; Arora et al., 2019), whose RKHS norm is bounded by $R_{\beta}$. For large $R_{\beta}$ and $m$, $\mathcal{F}_{\beta,m}$ represents a rich class of functions. Additionally, functions within $\mathcal{F}^{\mathcal{P}}_{\beta,m}$ can be viewed as the mean-embedding of the joint state-action pair onto the RKHS space (Muandet et al., 2016; Song et al., 2009; Smola et al., 2007). Below, we make one technical assumption which states that $\mathcal{F}^{\mathcal{P}}_{\theta,m_{Q}}$ is rich enough to represent the Q-function of all the policies within our policy class.

We define two mild conditions stating (1) boundedness of reward, and (2) regularity of the joint state and the policy.

We remark that the condition in Assumption 2 holds whenever ${\mathcal{S}}$ is bounded and $\mathrm{d}_{\mathcal{S}}\times\pi$ has an upper bounded density.

We measure the progress of MF-PPO in Algorithm 1 using the expected total reward

where $\nu^{*}$ is the stationary state distribution of the optimal policy $\pi^{*}$. We also denote $\sigma^{*}$ as the stationary state-action distribution induced by $\pi^{*}$. Note that we have:

for any policy $\pi$. Our main results are presented in the following theorem, showing that $\mathcal{L}(\pi_{k})$ converges to $\mathcal{L}(\pi^{*})$ at a sub-linear rate.

Theorem 4.1 states that, given sufficiently many agents and a large enough actor/critic network, MF-PPO attains global optimality at a sublinear rate. Our result shows that when solving the mean-field MDP, having more agents serves as a blessing instead of a curse. In addition, as will be demonstrated in our proof sketch, there exists an inherent phase transition depending on the number of agents, where the final optimality gap is dominated by statistical error for a small number of agents (first phase); and by optimization error for a large number of agents (second phase).

The complete proof of Theorem 4.1 takes a careful analysis on the error from policy evaluation (2) and the improvement step (3). The analysis on the outer iterations of MF-PPO can be overviewed as approximate mirror descent, which needs to take into account how the evaluation and improvement error interacts. Intuitively, the tuple $(\varepsilon_{k},\varepsilon^{\prime}_{k})$ describes the the effect of policy update when using approximate policy evaluation and policy improvement, and will be further clarified in the ensuing discussion. We present here the skeleton of our proof, and defer the technical detail to the appendix.

Proof Sketch. We first establish the global convergence of the policy evaluation (2) and improvement step (3).

Lemma 4.1 and 4.2 show that despite non-convexity, both the policy evaluation and the policy improvement steps converge approximately to the global optimal solution. In particular, given networks with large width, for a small number of iterations $T$, the optimization error $\mathcal{O}\left(T^{-1/2}\right)$ dominates the optimality gap; for a large number of iterations $T$, the statistical error $\mathcal{O}\left(N^{-1/2}\right)$ dominates the optimality gap.

With Lemma 4.1 and 4.2, we illustrate the main argument for the proof of Theorem 4.1. Let us assume the ideal case when $\epsilon_{k}=\epsilon^{\prime}_{k+1}=0$. Note that for $\epsilon_{k}=0$, we obtain the exact $Q$-function of policy $\pi_{k}$. For $\epsilon^{\prime}_{k+1}=0$, we obtain the ideal energy-based updated policy defined in Proposition 3.2. That is,

Without function approximation, problem (5) can be solved by treating each joint state $(s,\mathrm{d}_{\mathcal{S}})$ independently, hence one can apply the well known three-point lemma in mirror descent (Chen and Teboulle, 1993) and obtain that, for all $(s,\mathrm{d}_{\mathcal{S}})\in{\mathcal{S}}\times\mathcal{P}({\mathcal{S}})$:

From Lemma 6.1 in Kakade and Langford (2002), the expectation of the left hand side yields exactly $(1-\gamma)\left\{\mathcal{L}(\pi^{*})-\mathcal{L}(\pi_{k})\right\}$. Hence we have

Pinsker’s Inequality

combined with the observation $\|Q^{\pi_{k}}(s,\mathrm{d}_{\mathcal{S}},\cdot)\|_{\infty}\leq\overline{r}$, and basic inequality $-ax^{2}+bx\leq b^{2}/(4a)$ for $a>0$ gives us

By setting $\upsilon_{k}=\mathcal{O}(\sqrt{K})$, and telescoping the above inequality from $k=0$ to $K-1$, we obtain:

Note that the key element in the global convergence of MF-PPO is the recursion defined in (6), which holds whenever we have the exact $Q$-function of current policy and no function approximation is used when updating the next policy. Now MF-PPO conducts approximate policy evaluation ($\epsilon_{k}>0$), and after obtaining the approximate $Q$-function, conducts approximate policy improvement step ($\epsilon^{\prime}_{k+1}>0$). In addition, the error of approximating the $Q$-function introduced in the evaluation step can be further compounded in the improvement step. Nevertheless, recursion (6) still holds approximately, with additional terms representing the policy evaluation/improvement errors.

Finally, from Lemma 4.3, by telescoping inequality (8) from $k=0$ to $K-1$, we complete the proof of Theorem 4.1.

We perform experiments on the benchmark multi-agent particle environment (MPE) used in prior work (Lowe et al., 2017; Mordatch and Abbeel, 2018; Liu et al., 2019b). In the cooperative navigation task, $N$ agents each with position $x_{i}\in\mathbb{R}^{2}$ must move to cover $N$ fixed landmarks at positions $y_{i}\in\mathbb{R}^{2}$. They receive a team reward $R=-\sum_{i=1}^{N}\min_{j\in[N]}\lVert y_{i}-x_{j}\rVert_{2};$ In the cooperative push task, $N$ agents with position $x_{i}\in\mathbb{R}^{2}$ work together to push a ball $x\in\mathbb{R}^{2}$ to a fixed landmark $y\in\mathbb{R}^{2}$. They receive a team reward $R=-\min_{j\in[N]}\lVert x_{j}-x\rVert_{2}-\|x-y\|_{2}.$ Both tasks involve homogeneous agents.