Multi-Agent congestion cost minimization with linear function approximation

This work considers multiple agents traversing a network from a source node to the goal node. The cost to an agent for traveling a link has a private as well as a congestion component. The agent's objective is to find a path to the goal node with minimum overall cost in a decentralized way. We model this as a fully decentralized multi-agent reinforcement learning problem and propose a novel multi-agent congestion cost minimization (MACCM) algorithm. Our MACCM algorithm uses linear function approximations of transition probabilities and the global cost function. In the absence of a central controller and to preserve privacy, agents communicate the cost function parameters to their neighbors via a time-varying communication network. Moreover, each agent maintains its estimate of the global state-action value, which is updated via a multi-agent extended value iteration (MAEVI) sub-routine. We show that our MACCM algorithm achieves a sub-linear regret. The proof requires the convergence of cost function parameters, the MAEVI algorithm, and analysis of the regret bounds induced by the MAEVI triggering condition for each agent. We implement our algorithm on a two node network with multiple links to validate it. We first identify the optimal policy, the optimal number of agents going to the goal node in each period. We observe that the average regret is close to zero for 2 and 3 agents. The optimal policy captures the trade-off between the minimum cost of staying at a node and the congestion cost of going to the goal node. Our work is a generalization of learning the stochastic shortest path problem.

Multi-agent Natural Actor-critic Reinforcement Learning Algorithms

Both single-agent and multi-agent actor-critic algorithms are an important class of Reinforcement Learning algorithms. In this work, we propose three fully decentralized multi-agent natural actor-critic (MAN) algorithms. The agents' objective is to collectively learn a joint policy that maximizes the sum of averaged long-term returns of these agents. In the absence of a central controller, agents communicate the information to their neighbors via a time-varying communication network while preserving privacy. We prove the convergence of all the 3 MAN algorithms to a globally asymptotically stable point of the ODE corresponding to the actor update; these use linear function approximations. We use the Fisher information matrix to obtain the natural gradients. The Fisher information matrix captures the curvature of the Kullback-Leibler (KL) divergence between polices at successive iterates. We also show that the gradient of this KL divergence between policies of successive iterates is proportional to the objective function's gradient. Our MAN algorithms indeed use this \emph{representation} of the objective function's gradient. Under certain conditions on the Fisher information matrix, we prove that at each iterate, the optimal value via MAN algorithms can be better than that of the multi-agent actor-critic (MAAC) algorithm using the standard gradients. To validate the usefulness of our proposed algorithms, we implement all the 3 MAN algorithms on a bi-lane traffic network to reduce the average network congestion. We observe an almost 25% reduction in the average congestion in 2 MAN algorithms; the average congestion in another MAN algorithm is on par with the MAAC algorithm. We also consider a generic 15 agent MARL; the performance of the MAN algorithms is again as good as the MAAC algorithm. We attribute the better performance of the MAN algorithms to their use of the above representation.

On Feature Interactions Identified by Shapley Values of Binary Classification Games

For feature selection and related problems, we introduce the notion of classification game, a cooperative game, with features as players and hinge loss based characteristic function and relate a feature's contribution to Shapley value based error apportioning (SVEA) of total training error. Our major contribution is ($\star$) to show that for any dataset the threshold 0 on SVEA value identifies feature subset whose joint interactions for label prediction is significant or those features that span a subspace where the data is predominantly lying. In addition, our scheme ($\star$) identifies the features on which Bayes classifier doesn't depend but any surrogate loss function based finite sample classifier does; this contributes to the excess $0$-$1$ risk of such a classifier, ($\star$) estimates unknown true hinge risk of a feature, and ($\star$) relate the stability property of an allocation and negative valued SVEA by designing the analogue of core of classification game. Due to Shapley value's computationally expensive nature, we build on a known Monte Carlo based approximation algorithm that computes characteristic function (Linear Programs) only when needed. We address the potential sample bias problem in feature selection by providing interval estimates for SVEA values obtained from multiple sub-samples. We illustrate all the above aspects on various synthetic and real datasets and show that our scheme achieves better results than existing recursive feature elimination technique and ReliefF in most cases. Our theoretically grounded classification game in terms of well defined characteristic function offers interpretability and explainability of our framework, including identification of important features.