Joshua R. Bertram

Joshua R Bertram, Peng Wei, Joseph Zambreno

Pre-departure flight plan scheduling for Urban Air Mobility (UAM) and cargo delivery drones will require on-demand scheduling of large numbers of aircraft. We examine the scalability of an algorithm known as FastMDP which was shown to perform well in deconflicting many dozens of aircraft in a dense airspace environment with terrain. We show that the algorithm can adapted to perform first-come-first-served pre-departure flight plan scheduling where conflict free flight plans are generated on demand. We demonstrate a parallelized implementation of the algorithm on a Graphics Processor Unit (GPU) which we term FastMDP-GPU and show the level of performance and scaling that can be achieved. Our results show that on commodity GPU hardware we can perform flight plan scheduling against 2000-3000 known flight plans and with server-class hardware the performance can be higher. We believe the results show promise for implementing a large scale UAM scheduler capable of performing on-demand flight scheduling that would be suitable for both a centralized or distributed flight planning system

Joshua R. Bertram, Peng Wei

We present a method for pursuit/evasion that is highly efficient and and scales to large teams of aircraft. The underlying algorithm is an efficient algorithm for solving Markov Decision Processes (MDPs) that supports fully continuous state spaces. We demonstrate the algorithm in a team pursuit/evasion setting in a 3D environment using a pseudo-6DOF model and study performance by varying sizes of team members. We show that as the number of aircraft in the simulation grows, computational performance remains efficient and is suitable for real-time systems. We also define probability-to-win and survivability metrics that describe the teams' performance over multiple trials, and show that the algorithm performs consistently. We provide numerical results showing control inputs for a typical 1v1 encounter and provide videos for 1v1, 2v2, 3v3, 4v4, and 10v10 contests to demonstrate the ability of the algorithm to adapt seamlessly to complex environments.

Marc Brittain, Josh Bertram, Xuxi Yang et al.

Experience replay is widely used in deep reinforcement learning algorithms and allows agents to remember and learn from experiences from the past. In an effort to learn more efficiently, researchers proposed prioritized experience replay (PER) which samples important transitions more frequently. In this paper, we propose Prioritized Sequence Experience Replay (PSER) a framework for prioritizing sequences of experience in an attempt to both learn more efficiently and to obtain better performance. We compare the performance of PER and PSER sampling techniques in a tabular Q-learning environment and in DQN on the Atari 2600 benchmark. We prove theoretically that PSER is guaranteed to converge faster than PER and empirically show PSER substantially improves upon PER.

Josh Bertram, Peng Wei

We present a method for a certain class of Markov Decision Processes (MDPs) that can relate the optimal policy back to one or more reward sources in the environment. For a given initial state, without fully computing the value function, q-value function, or the optimal policy the algorithm can determine which rewards will and will not be collected, whether a given reward will be collected only once or continuously, and which local maximum within the value function the initial state will ultimately lead to. We demonstrate that the method can be used to map the state space to identify regions that are dominated by one reward source and can fully analyze the state space to explain all actions. We provide a mathematical framework to show how all of this is possible without first computing the optimal policy or value function.

Joshua R. Bertram, Peng Wei

We propose an algorithm for deterministic continuous Markov Decision Processes with sparse rewards that computes the optimal policy exactly with no dependency on the size of the state space. The algorithm has time complexity of $O( |R|^3 \times |A|^2 )$ and memory complexity of $O( |R| \times |A| )$, where $|R|$ is the number of reward sources and $|A|$ is the number of actions. Furthermore, we describe a companion algorithm that can follow the optimal policy from any initial state without computing the entire value function, instead computing on-demand the value of states as they are needed. The algorithm to solve the MDP does not depend on the size of the state space for either time or memory complexity, and the ability to follow the optimal policy is linear in time and space with the path length of following the optimal policy from the initial state. We demonstrate the algorithm operation side by side with value iteration on tractable MDPs.

Joshua R. Bertram, Xuxi Yang, Peng Wei

Markov Decision Processes (MDPs) are a mathematical framework for modeling sequential decision making under uncertainty. The classical approaches for solving MDPs are well known and have been widely studied, some of which rely on approximation techniques to solve MDPs with large state space and/or action space. However, most of these classical solution approaches and their approximation techniques still take much computation time to converge and usually must be re-computed if the reward function is changed. This paper introduces a novel alternative approach for exactly and efficiently solving deterministic, continuous MDPs with sparse reward sources. When the environment is such that the "distance" between states can be determined in constant time, e.g. grid world, our algorithm offers $O( |R|^2 \times |A|^2 \times |S|)$, where $|R|$ is the number of reward sources, $|A|$ is the number of actions, and $|S|$ is the number of states. Memory complexity for the algorithm is $O( |S| + |R| \times |A|)$. This new approach opens new avenues for boosting computational performance for certain classes of MDPs and is of tremendous value for MDP applications such as robotics and unmanned systems. This paper describes the algorithm and presents numerical experiment results to demonstrate its powerful computational performance. We also provide rigorous mathematical description of the approach.