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This paper considers the problem of cooperative localization of multiple robots under uncertainty, communicating over a partially connected, dynamic communication network and assisted by an agile landmark. Each robot owns an IMU and a relative pose sensing suite, which can get faulty due to system or environmental uncertainty, and therefore exhibit large bias in their estimation output. For the robots to localize accurately under sensor failure and system or environmental uncertainty, a novel Distributed Learning based Decentralized Cooperative Localization (DL-DCL) algorithm is proposed that involves real-time learning of an information fusion strategy by each robot for combining pose estimates from its own sensors as well as from those of its neighboring robots, and utilizing the moving landmark's pose information as a feedback to the learning process. Convergence analysis shows that the learning process converges exponentially under certain reasonable assumptions. Simulations involving sensor failures inducing around 40-60 times increase in the nominal bias show DL-DCL's estimation performance to be approximately 40% better than the well-known covariance-based estimate fusion methods. For the evaluation of DL-DCL's implementability and fault-tolerance capability in practice, a high-fidelity simulation is carried out in Gazebo with ROS2.