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Register a new userA General Framework for Learning Mean-Field Games
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Renyuan Xu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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This paper presents a general mean-field game (GMFG) framework for simultaneous learning and decision-making in stochastic games with a large population. It first establishes the existence of a unique Nash Equilibrium to this GMFG, and demonstrates that naively combining Q-learning with the fixed-point approach in classical MFGs yields unstable algorithms. It then proposes value-based and policy-based reinforcement learning algorithms (GMF-P and GMF-P respectively) with smoothed policies, with analysis of convergence property and computational complexity. The experiments on repeated Ad auction problems demonstrate that GMF-V-Q, a specific GMF-V algorithm based on Q-learning, is efficient and robust in terms of convergence and learning accuracy. Moreover, its performance is superior in convergence, stability, and learning ability, when compared with existing algorithms for multi-agent reinforcement learning.
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Multi-agent reinforcement learning (MARL), despite its popularity and empirical success, suffers from the curse of dimensionality. This paper builds the mathematical framework to approximate cooperative MARL by a mean-field control (MFC) approach, and shows that the approximation error is of $mathcal{O}(frac{1}{sqrt{N}})$. By establishing an appropriate form of the dynamic programming principle for both the value function and the Q function, it proposes a model-free kernel-based Q-learning algorithm (MFC-K-Q), which is shown to have a linear convergence rate for the MFC problem, the first of its kind in the MARL literature. It further establishes that the convergence rate and the sample complexity of MFC-K-Q are independent of the number of agents $N$, which provides an $mathcal{O}(frac{1}{sqrt{N}})$ approximation to the MARL problem with $N$ agents in the learning environment. Empirical studies for the network traffic congestion problem demonstrate that MFC-K-Q outperforms existing MARL algorithms when $N$ is large, for instance when $N>50$.
In this paper, we develop a PDE approach to consider the optimal strategy of mean field controlled stochastic system. Firstly, we discuss mean field SDEs and associated Fokker-Plank eqautions. Secondly, we consider a fully-coupled system of forward-backward PDEs. The backward one is the Hamilton-Jacobi-Bellman equation while the forward one is the Fokker-Planck equation. Our main result is to show the existence of classical solutions of the forward-backward PDEs in the class $H^{1+frac{1}{4},2+frac{1}{2}}([0,T]timesmathbb{R}^n)$ by use of the Schauder fixed point theorem. Then, we use the solution to give the optimal strategy of the mean field stochastic control problem. Finally, we give an example to illustrate the role of our main result.
Entropy regularization has been extensively adopted to improve the efficiency, the stability, and the convergence of algorithms in reinforcement learning. This paper analyzes both quantitatively and qualitatively the impact of entropy regularization for Mean Field Game (MFG) with learning in a finite time horizon. Our study provides a theoretical justification that entropy regularization yields time-dependent policies and, furthermore, helps stabilizing and accelerating convergence to the game equilibrium. In addition, this study leads to a policy-gradient algorithm for exploration in MFG. Under this algorithm, agents are able to learn the optimal exploration scheduling, with stable and fast convergence to the game equilibrium.
We study the training of regularized neural networks where the regularizer can be non-smooth and non-convex. We propose a unified framework for stochastic proximal gradient descent, which we term ProxGen, that allows for arbitrary positive preconditioners and lower semi-continuous regularizers. Our framework encompasses standard stochastic proximal gradient methods without preconditioners as special cases, which have been extensively studied in various settings. Not only that, we present two important update rules beyond the well-known standard methods as a byproduct of our approach: (i) the first closed-form proximal mappings of $ell_q$ regularization ($0 leq q leq 1$) for adaptive stochastic gradient methods, and (ii) a revised version of ProxQuant that fixes a caveat of the original approach for quantization-specific regularizers. We analyze the convergence of ProxGen and show that the whole family of ProxGen enjoys the same convergence rate as stochastic proximal gradient descent without preconditioners. We also empirically show the superiority of proximal methods compared to subgradient-based approaches via extensive experiments. Interestingly, our results indicate that proximal methods with non-convex regularizers are more effective than those with convex regularizers.
One of the challenges for multi-agent reinforcement learning (MARL) is designing efficient learning algorithms for a large system in which each agent has only limited or partial information of the entire system. In this system, it is desirable to learn policies of a decentralized type. A recent and promising paradigm to analyze such decentralized MARL is to take network structures into consideration. While exciting progress has been made to analyze decentralized MARL with the network of agents, often found in social networks and team video games, little is known theoretically for decentralized MARL with the network of states, frequently used for modeling self-driving vehicles, ride-sharing, and data and traffic routing. This paper proposes a framework called localized training and decentralized execution to study MARL with network of states, with homogeneous (a.k.a. mean-field type) agents. Localized training means that agents only need to collect local information in their neighboring states during the training phase; decentralized execution implies that, after the training stage, agents can execute the learned decentralized policies, which only requires knowledge of the agents current states. The key idea is to utilize the homogeneity of agents and regroup them according to their states, thus the formulation of a networked Markov decision process with teams of agents, enabling the update of the Q-function in a localized fashion. In order to design an efficient and scalable reinforcement learning algorithm under such a framework, we adopt the actor-critic approach with over-parameterized neural networks, and establish the convergence and sample complexity for our algorithm, shown to be scalable with respect to the size of both agents and states.
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