We propose a formulation of the stochastic cutting stock problem as a discounted infinite-horizon Markov decision process. At each decision epoch, given current inventory of items, an agent chooses in which patterns to cut objects in stock in anticipation of the unknown demand. An optimal solution corresponds to a policy that associates each state with a decision and minimizes the expected total cost. Since exact algorithms scale exponentially with the state-space dimension, we develop a heuristic solution approach based on reinforcement learning. We propose an approximate policy iteration algorithm in which we apply a linear model to approximate the action-value function of a policy. Policy evaluation is performed by solving the projected Bellman equation from a sample of state transitions, decisions and costs obtained by simulation. Due to the large decision space, policy improvement is performed via the cross-entropy method. Computational experiments are carried out with the use of realistic data to illustrate the application of the algorithm. Heuristic policies obtained with polynomial and Fourier basis functions are compared with myopic and random policies. Results indicate the possibility of obtaining policies capable of adequately controlling inventories with an average cost up to 80% lower than the cost obtained by a myopic policy.
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Optimal control of general nonlinear systems is a central challenge in automation. Data-driven approaches to control, enabled by powerful function approximators, have recently had great success in tackling challenging robotic applications. However, such methods often obscure the structure of dynamics and control behind black-box over-parameterized representations, thus limiting our ability to understand the closed-loop behavior. This paper adopts a hybrid-system view of nonlinear modeling and control that lends an explicit hierarchical structure to the problem and breaks down complex dynamics into simpler localized units. Therefore, we consider a sequence modeling paradigm that captures the temporal structure of the data and derive an expecation-maximization (EM) algorithm that automatically decomposes nonlinear dynamics into stochastic piecewise affine dynamical systems with nonlinear boundaries. Furthermore, we show that these time-series models naturally admit a closed-loop extension that we use to extract locally linear or polynomial feedback controllers from nonlinear experts via imitation learning. Finally, we introduce a novel hybrid realtive entropy policy search (Hb-REPS) technique that incorporates the hierarchical nature of hybrid systems and optimizes a set of time-invariant local feedback controllers derived from a locally polynomial approximation of a global value function.
In this paper, we study a new stochastic submodular maximization problem with state-dependent costs and rejections. The input of our problem is a budget constraint $B$, and a set of items whose states (i.e., the marginal contribution and the cost of an item) are drawn from a known probability distribution. The only way to know the realized state of an item is to probe that item. We allow rejections, i.e., after probing an item and knowing its actual state, we must decide immediately and irrevocably whether to add that item to our solution or not. Our objective is to sequentially probe/selet a best group of items subject to a budget constraint on the total cost of the selected items. We present a constant approximate solution to this problem. We show that our solution can be extended to an online setting.
Reinforcement Learning (RL) has been shown effective in domains where the agent can learn policies by actively interacting with its operating environment. However, if we change the RL scheme to offline setting where the agent can only update its policy via static datasets, one of the major issues in offline reinforcement learning emerges, i.e. distributional shift. We propose a Pessimistic Offline Reinforcement Learning (PessORL) algorithm to actively lead the agent back to the area where it is familiar by manipulating the value function. We focus on problems caused by out-of-distribution (OOD) states, and deliberately penalize high values at states that are absent in the training dataset, so that the learned pessimistic value function lower bounds the true value anywhere within the state space. We evaluate the PessORL algorithm on various benchmark tasks, where we show that our method gains better performance by explicitly handling OOD states, when compared to those methods merely considering OOD actions.
We study constrained reinforcement learning (CRL) from a novel perspective by setting constraints directly on state density functions, rather than the value functions considered by previous works. State density has a clear physical and mathematical interpretation, and is able to express a wide variety of constraints such as resource limits and safety requirements. Density constraints can also avoid the time-consuming process of designing and tuning cost functions required by value function-based constraints to encode system specifications. We leverage the duality between density functions and Q functions to develop an effective algorithm to solve the density constrained RL problem optimally and the constrains are guaranteed to be satisfied. We prove that the proposed algorithm converges to a near-optimal solution with a bounded error even when the policy update is imperfect. We use a set of comprehensive experiments to demonstrate the advantages of our approach over state-of-the-art CRL methods, with a wide range of density constrained tasks as well as standard CRL benchmarks such as Safety-Gym.
Deep reinforcement learning suggests the promise of fully automated learning of robotic control policies that directly map sensory inputs to low-level actions. However, applying deep reinforcement learning methods on real-world robots is exceptionally difficult, due both to the sample complexity and, just as importantly, the sensitivity of such methods to hyperparameters. While hyperparameter tuning can be performed in parallel in simulated domains, it is usually impractical to tune hyperparameters directly on real-world robotic platforms, especially legged platforms like quadrupedal robots that can be damaged through extensive trial-and-error learning. In this paper, we develop a stable variant of the soft actor-critic deep reinforcement learning algorithm that requires minimal hyperparameter tuning, while also requiring only a modest number of trials to learn multilayer neural network policies. This algorithm is based on the framework of maximum entropy reinforcement learning, and automatically trades off exploration against exploitation by dynamically and automatically tuning a temperature parameter that determines the stochasticity of the policy. We show that this method achieves state-of-the-art performance on four standard benchmark environments. We then demonstrate that it can be used to learn quadrupedal locomotion gaits on a real-world Minitaur robot, learning to walk from scratch directly in the real world in two hours of training.
Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.
We present an end-to-end framework for solving the Vehicle Routing Problem (VRP) using reinforcement learning. In this approach, we train a single model that finds near-optimal solutions for problem instances sampled from a given distribution, only by observing the reward signals and following feasibility rules. Our model represents a parameterized stochastic policy, and by applying a policy gradient algorithm to optimize its parameters, the trained model produces the solution as a sequence of consecutive actions in real time, without the need to re-train for every new problem instance. On capacitated VRP, our approach outperforms classical heuristics and Google's OR-Tools on medium-sized instances in solution quality with comparable computation time (after training). We demonstrate how our approach can handle problems with split delivery and explore the effect of such deliveries on the solution quality. Our proposed framework can be applied to other variants of the VRP such as the stochastic VRP, and has the potential to be applied more generally to combinatorial optimization problems.
This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.
This paper presents a safety-aware learning framework that employs an adaptive model learning method together with barrier certificates for systems with possibly nonstationary agent dynamics. To extract the dynamic structure of the model, we use a sparse optimization technique, and the resulting model will be used in combination with control barrier certificates which constrain feedback controllers only when safety is about to be violated. Under some mild assumptions, solutions to the constrained feedback-controller optimization are guaranteed to be globally optimal, and the monotonic improvement of a feedback controller is thus ensured. In addition, we reformulate the (action-)value function approximation to make any kernel-based nonlinear function estimation method applicable. We then employ a state-of-the-art kernel adaptive filtering technique for the (action-)value function approximation. The resulting framework is verified experimentally on a brushbot, whose dynamics is unknown and highly complex.
Modern communication networks have become very complicated and highly dynamic, which makes them hard to model, predict and control. In this paper, we develop a novel experience-driven approach that can learn to well control a communication network from its own experience rather than an accurate mathematical model, just as a human learns a new skill (such as driving, swimming, etc). Specifically, we, for the first time, propose to leverage emerging Deep Reinforcement Learning (DRL) for enabling model-free control in communication networks; and present a novel and highly effective DRL-based control framework, DRL-TE, for a fundamental networking problem: Traffic Engineering (TE). The proposed framework maximizes a widely-used utility function by jointly learning network environment and its dynamics, and making decisions under the guidance of powerful Deep Neural Networks (DNNs). We propose two new techniques, TE-aware exploration and actor-critic-based prioritized experience replay, to optimize the general DRL framework particularly for TE. To validate and evaluate the proposed framework, we implemented it in ns-3, and tested it comprehensively with both representative and randomly generated network topologies. Extensive packet-level simulation results show that 1) compared to several widely-used baseline methods, DRL-TE significantly reduces end-to-end delay and consistently improves the network utility, while offering better or comparable throughput; 2) DRL-TE is robust to network changes; and 3) DRL-TE consistently outperforms a state-ofthe-art DRL method (for continuous control), Deep Deterministic Policy Gradient (DDPG), which, however, does not offer satisfying performance.
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