We introduce the problem of Dynamic Real-time Multimodal Routing (DREAMR), which requires planning and executing routes under uncertainty for an autonomous agent. The agent has access to a time-varying transit vehicle network in which it can use multiple modes of transportation. For instance, a drone can either fly or ride on terrain vehicles for segments of their routes. DREAMR is a difficult problem of sequential decision making under uncertainty with both discrete and continuous variables. We design a novel hierarchical hybrid planning framework to solve the DREAMR problem that exploits its structural decomposability. Our framework consists of a global open-loop planning layer that invokes and monitors a local closed-loop execution layer. Additional abstractions allow efficient and seamless interleaving of planning and execution. We create a large-scale simulation for DREAMR problems, with each scenario having hundreds of transportation routes and thousands of connection points. Our algorithmic framework significantly outperforms a receding horizon control baseline, in terms of elapsed time to reach the destination and energy expended by the agent.
A Markov Decision Process or MDP @cite_33 is defined by @math , where and are the system's state and action spaces, @math is the transition function, where @math and @math , @math is the reward function, and @math the discount factor. Solving an MDP yields a policy @math which maximizes the or the expected reward-to-go from each state. An MDP can be solved by , a dynamic programming (DP) method that computes the optimal value function @math . We obtain a single @math for infinite horizon problems and @math ( @math is the maximum number of timesteps) for finite horizon or episodic problems. For large or continuous spaces, we can approximate the value function locally with multilinear interpolation @cite_29 , or globally with basis functions @cite_10 . Our framework uses approximate DP extensively @cite_6 .
@cite_29 @cite_10 @cite_33 @cite_6
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Consider the problem of approximating the optimal policy of a Markov decision process (MDP) by sampling state transitions. In contrast to existing reinforcement learning methods that are based on successive approximations to the nonlinear Bellman equation, we propose a Primal-Dual @math Learning method in light of the linear duality between the value and policy. The @math learning method is model-free and makes primal-dual updates to the policy and value vectors as new data are revealed. For infinite-horizon undiscounted Markov decision process with finite state space @math and finite action space @math , the @math learning method finds an @math -optimal policy using the following number of sample transitions @math where @math is an upper bound of mixing times across all policies and @math is a parameter characterizing the range of stationary distributions across policies. The @math learning method also applies to the computational problem of MDP where the transition probabilities and rewards are explicitly given as the input. In the case where each state transition can be sampled in @math time, the @math learning method gives a sublinear-time algorithm for solving the averaged-reward MDP. Consider a Markov decision process (MDP) that admits a set of state-action features, which can linearly express the process's probabilistic transition model. We propose a parametric Q-learning algorithm that finds an approximate-optimal policy using a sample size proportional to the feature dimension @math and invariant with respect to the size of the state space. To further improve its sample efficiency, we exploit the monotonicity property and intrinsic noise structure of the Bellman operator, provided the existence of anchor state-actions that imply implicit non-negativity in the feature space. We augment the algorithm using techniques of variance reduction, monotonicity preservation, and confidence bounds. It is proved to find a policy which is @math -optimal from any initial state with high probability using @math sample transitions for arbitrarily large-scale MDP with a discount factor @math . A matching information-theoretical lower bound is proved, confirming the sample optimality of the proposed method with respect to all parameters (up to polylog factors). We consider the problem of learning an unknown Markov Decision Process (MDP) that is weakly communicating in the infinite horizon setting. We propose a Thompson Sampling-based reinforcement learning algorithm with dynamic episodes (TSDE). At the beginning of each episode, the algorithm generates a sample from the posterior distribution over the unknown model parameters. It then follows the optimal stationary policy for the sampled model for the rest of the episode. The duration of each episode is dynamically determined by two stopping criteria. The first stopping criterion controls the growth rate of episode length. The second stopping criterion happens when the number of visits to any state-action pair is doubled. We establish @math bounds on expected regret under a Bayesian setting, where @math and @math are the sizes of the state and action spaces, @math is time, and @math is the bound of the span. This regret bound matches the best available bound for weakly communicating MDPs. Numerical results show it to perform better than existing algorithms for infinite horizon MDPs. In this paper we consider the problem of computing an ϵ-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in O(1) time. Given such a DMDP with states , actions , discount factor γ∈(0,1), and rewards in range [0,1] we provide an algorithm which computes an ϵ-optimal policy with probability 1−δ where both the run time spent and number of sample taken is upper bounded by O[ | || | (1−γ)3ϵ2 log( | || | (1−γ)δϵ )log( 1 (1−γ)ϵ )] . For fixed values of ϵ, this improves upon the previous best known bounds by a factor of (1−γ)−1 and matches the sample complexity lower bounds proved in azar2013minimax up to logarithmic factors. We also extend our method to computing ϵ-optimal policies for finite-horizon MDP with a generative model and provide a nearly matching sample complexity lower bound.