Logarithmic Regret for Linear Markov Decision Processes with Adversarial Corruptions

We initiate the study of episodic reinforcement learning (RL) under adversarial corruptions in both the rewards and the transition probabilities of the underlying system, extending recent results for the special case of multiarmed bandits. We provide a framework that modifies the aggressive exploration enjoyed by existing reinforcement learning approaches based on optimism in the face of uncertainty by complementing them with principles from action elimination. Importantly, our framework circumvents the major challenges posed by naively applying action elimination in the RL setting, as formalized by a lower bound we demonstrate. Our framework yields efficient algorithms that (a) attain near-optimal regret in the absence of corruptions and (b) adapt to unknown levels of corruption, enjoying regret guarantees that degrade gracefully in the total corruption encountered. To showcase the generality of our approach, we derive results for both tabular settings (where states and actions are finite) and linear Markov decision process settings (where the dynamics and rewards admit a linear underlying representation). Notably, our work provides the first sublinear regret guarantee that accommodates any deviation from purely independent and identically distributed transitions in the bandit-feedback model for episodic reinforcement learning. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2021.0202 .

In reinforcement learning, when the state space is enormous or infinite, it is not feasible to find the exact value for each state in the memory. A common way to tackle this problem is to adopt linear value function approximation technique. In this paper, we review some commonly used linear value function approximation algorithms, which divided into two classes. The first class is the so-called on-policy algorithms that mainly include temporal difference and least-squares-based methods. The second is the off-policy algorithms that use two strategies to solve the exploration-exploitation problem. From the perspective of algorithm design, on-policy is simple, but it does not take both exploration and exploitation into consideration when collecting data, while the off-policy can well balance the relation between exploration and exploitation. In the end, we first present some experiments to demonstrate the usefulness of linear value function approximation technique in the continuous and high dimensional scenes, and then in-class and out-class experiments are respectively conducted to further illustrate the performance of different approximation algorithms.KeywordsReinforcement learningLinear value function approximationLeast-squares temporal difference

Data Stream Processing (DSP) has emerged as a key enabler to develop pervasive services that require to process data in a near real-time fashion. DSP applications keep up with the high volume of produced data by scaling their execution on multiple computing nodes, so as to process the incoming data flow in parallel. Workloads variability requires to elastically adapt the application parallelism at run-time in order to avoid over-provisioning. Elasticity policies for DSP have been widely investigated, but mostly under the simplifying assumption of homogeneous infrastructures. The resulting solutions do not capture the richness and inherent complexity of modern infrastructures, where heterogeneous computing resources are available on-demand. In this paper, we formulate the problem of controlling elasticity on heterogeneous resources as a Markov Decision Process (MDP). The resulting MDP is not easily solved by traditional techniques due to state space explosion, and thus we show how linear Function Approximation and Tile Coding can be used to efficiently compute elasticity policies at run-time. In order to deal with parameters uncertainty, we integrate the proposed approach with Reinforcement Learning algorithms. Our numerical evaluation shows the efficacy of the presented solutions compared to standard methods in terms of accuracy and convergence speed.

Introduction Reinforcement learning formalizes the concept of learning from interactions.1 Broadly, reinforcement learning focuses on a setting in which an agent (decision maker) sequentially interacts with an environment that is partially unknown to them. At each stage, the agent takes an action and receives a reward. The objective of the agent is to maximize rewards accumulated in the long run. There are many situations in health care where decisions are made sequentially for which reinforcement learning approaches could prove useful for decision making. Throughout this article, we consider treatment prescription as an archetypical example to connect reinforcement learning concepts to a health care setting. In this setting, the care provider, the prescribed treatment, and the patients can be viewed as the agent, the action, and the environment, respectively, as depicted in Figure 1.Figure 1: Sequential treatment of AKI or CKD complications modeled as a reinforcement learning problem.Background In this section, with the objective of making reinforcement learning literature more accessible to a clinical audience, we briefly introduce related fundamental concepts and approaches. We refer the interested reader to Sutton and Barto1 for a comprehensive introduction to reinforcement learning. Markov Decision Processes Markov decision processes (MDPs) are a formalism of the sequential decision-making problem that has been central to the theoretical and practical advancements of reinforcement learning. In each stage of an MDP, the agent observes the state of the environment and takes an action, which, in turn, results in a change of the state. This change of state is assumed to be probabilistic with the next state being determined only by the preceding state, the chosen action, and the transition probability. The agent also receives a reward that is a function of the taken action, the preceding state, and the subsequent state. In an MDP, the objective of the agent is to maximize the return defined as the reward accumulated over a time horizon. In some applications, it is common to consider the horizon to be infinite, in which case the future rewards are discounted by a factor smaller than one. The selection of action by the agent on the basis of the observed state is known as the policy. More formally, a policy is a probabilistic mapping from states to each possible action. Because the policy and the reward are a function of the state, it is critical to estimate the utility of being in a certain state. More specifically, the value function is defined as the expected return starting from a given state under the chosen policy. Under this formalism, the objective of the agent is to find the optimal policy that maximizes the value function for all states. Reinforcement Learning Methods Action-value methods are a class of reinforcement learning methods in which the actions are chosen on the basis of the estimation of their long-term value. A prominent example of an action-value method is Q-learning in which the agent iteratively takes actions with the highest estimated values and updates the action-state value function on the basis of new observations. Policy gradient methods are another class of reinforcement learning methods that seek to optimize the policy directly instead of choosing actions on the basis of their respective estimated value. Such methods could be advantageous in health care applications that entail a large number of possible actions, e.g., when recommending a wide range of drug dosages or treatment options. Clinical Applications Reinforcement learning frameworks and methods are broadly applicable to clinical settings in which decisions are made sequentially. A prominent clinical application of reinforcement learning is for treatment recommendation, which has been studied across a variety of diseases and treatments including radiation and chemotherapy for cancer, brain stimulation for epilepsy, and treatment strategies for sepsis.2–5 In such treatment recommendation settings, a policy is commonly known as a dynamic treatment regime. There are various other clinical applications of reinforcement learning including diagnosis, medical imaging, and decision support tools (see refs. 2–5 and the references therein). Reinforcement Learning in Nephrology Although there have been recent applications of machine learning in nephrology,6,7 to the best of the authors' knowledge, the application of reinforcement learning to nephrology has been primarily limited to optimizing the erythropoietin dosage in hemodialysis patients.8,9 However, there are other settings where reinforcement learning has the potential to improve patient care in nephrology. For example, reinforcement learning methods can be adopted in the treatment of the complications of AKI or CKD (Figure 1). In this problem, the state models the conditions of the patient (e.g., vital signs, laboratory test results including urine and blood tests, and urine output measurements). The action refers to the treatment options (e.g., the dosage of medications such as sodium polystyrene sulfonate, and hemodialysis). The reward models the improvement in patient conditions. Similarly, reinforcement learning can help automate and optimize the dosage of immunosuppressive drugs in kidney transplants. Challenges and Opportunities Despite the success of reinforcement learning in several simplified clinical settings, their large-scale application to patient care faces several open challenges. The complexity of human biology complicates modeling clinical decision making as a reinforcement learning problem. The state space in such settings is often enormous, which could make a purely computational approach infeasible. Moreover, modeling all potential objectives a priori as a reward function may not be feasible. To overcome these challenges and realize the potential of reinforcement learning, clinical insight can play a pivotal role. More specifically, restricting the state space to only include highly relevant clinical variables could greatly reduce the computational complexity. Furthermore, using inverse reinforcement learning,2 relevant reward functions can be learned from retrospective studies assuming the optimality of clinical decisions. Another critical challenge is addressing moral and ethical concerns. It is imperative to ensure that reinforcement learning methods do not cause harm to the patient. To this end, there exists a need for a thorough validation of such methods before their use in patient care. Hence, there is a need to go beyond retrospective studies that have been used for the proof of concept of most existing reinforcement learning methods in health care applications.2,3 The lessons learned from the success of reinforcement learning in other application areas (e.g., self-driving cars) can help navigate the path to realizing its potential in health care. Accessible open-source simulation environments that enable researchers to compare various approaches are essential to the field of reinforcement learning. OpenAI Gym is currently the leading toolkit containing a wide range of simulated environments, e.g., surgical robotics.10 The development of high-quality and reliable simulation environments for nephrology and other health care applications can facilitate the development and validation of reinforcement learning methods beyond limited retrospective studies. The adoption of methods validated in such simulation environments in actual clinical settings will require clinicians' oversight. Similar to how self-driving cars require a human driver to ensure collision avoidance, clinicians' oversight is critical to ensure the safety of the patients, especially in the early stages of the adoption of reinforcement learning methods. The data from clinicians' decisions (e.g., overruling the automated treatment recommendation) can be used to improve the reliability of autonomous systems over time and reduce the burden of clinicians' oversight.

In computer science, reinforcement learning is a powerful framework with which artificial agents can learn to maximize their performance for any given Markov decision process (MDP). Advances over the last decade, in combination with deep neural networks, have enjoyed performance advantages over humans in many difficult task settings. However, such frameworks perform far less favorably when evaluated in their ability to generalize or transfer representations across different tasks. Existing algorithms that facilitate transfer typically are limited to cases in which the transition function or the optimal policy is portable to new contexts, but achieving “deep transfer” characteristic of human behavior has been elusive. Such transfer typically requires discovery of abstractions that permit analogical reuse of previously learned representations to superficially distinct tasks. Here, we demonstrate that abstractions that minimize error in predictions of reward outcomes generalize across tasks with different transition and reward functions. Such reward-predictive representations compress the state space of a task into a lower dimensional representation by combining states that are equivalent in terms of both the transition and reward functions. Because only state equivalences are considered, the resulting state representation is not tied to the transition and reward functions themselves and thus generalizes across tasks with different reward and transition functions. These results contrast with those using abstractions that myopically maximize reward in any given MDP and motivate further experiments in humans and animals to investigate if neural and cognitive systems involved in state representation perform abstractions that facilitate such equivalence relations.

This paper revisits the temporal difference (TD) learning algorithm for the policy evaluation tasks in reinforcement learning. Typically, the performance of TD(0) and TD( λ) is very sensitive to the choice of stepsizes. Oftentimes, TD(0) suffers from slow convergence. Motivated by the tight link between the TD(0) learning algorithm and the stochastic gradient methods, we develop a provably convergent adaptive projected variant of the TD(0) learning algorithm with linear function approximation that we term AdaTD(0). In contrast to the TD(0), AdaTD(0) is robust or less sensitive to the choice of stepsizes. Analytically, we establish that to reach an ϵ accuracy, the number of iterations needed is [Formula: see text] in the general case, where ρ represents the speed of the underlying Markov chain converges to the stationary distribution. This implies that the iteration complexity of AdaTD(0) is no worse than that of TD(0) in the worst case. When the stochastic semi-gradients are sparse, we provide theoretical acceleration of AdaTD(0). Going beyond TD(0), we develop an adaptive variant of TD( λ), which is referred to as AdaTD( λ). Empirically, we evaluate the performance of AdaTD(0) and AdaTD( λ) on several standard reinforcement learning tasks, which demonstrate the effectiveness of our new approaches.

Reinforcement Learning (RL) Markov Decision Processes is studied with an emphasis on the well-studied exploration problem. We first formulate and discuss a definition of “efficient” algorithms that is termed Probably Approximately Correct (PAC) in RL. Next we provide general sufficient conditions for such an algorithm that applies to several different modeling assumptions. The conditions can be used to demonstrate that efficient learning is possible in finite MDPs, with either a model-based or model-free approach, in factored MDPs, and in continuous MDPs with linear dynamics. In the reinforcement-learning (RL) problem (Sutton & Barto 1998), an agent acts in an unknown or incompletely known environment with the goal of maximizing an external reward signal. In the most standard mathematical formulation of the problem, the environment is modeled as a Markov Decision Process (MDP) and the goal of the agent is to obtain near-optimal discounted return. Over the years, many algorithms have been proposed for this problem, but analyses of their performances have been relatively scarce. In fact, until recently, most theoretical guarantees have been that certain algorithms will discover an optimal policy in the limit, after an infinite amount of experience. In contrast, several attempts have been made to study “Probably Approximately Correct” or PAC-MDP algorithms, which exhibit near-optimal behavior in polynomial time and experience. This paper discusses several extensions of those results. We present a theorem that provides sufficient conditions for an algorithm to be PAC-MDP. We examine these conditions and show how they can be applied to prove that efficient learning is possible in three interesting scenerios: finite MDPs (i.e. the “Tabular case”),

Reinforcement Learning (RL) is a machine learning methodology that develops the capability to make sequential decisions in intricate issues using trial-and-error techniques. RL has become increasingly prevalent for decision-making and control tasks in diverse fields such as industrial processes, biochemical systems and energy management. This review paper presents a comprehensive examination of the development, models, algorithms and practical uses of RL, with a specific emphasis on its application in process control. The study examines the fundamental theories, methodology and applications of RL, classifying them into two categories: classical RL such as such as Markov decision processes (MDP) and deep RL viz., actor critic methods. RL is a topic of discussion in multiple process industries, such as industrial chemical process control, biochemical process control, energy systems, wastewater treatment and the oil and gas sector. Nevertheless, the paper also highlights challenges that hinder its larger acceptance, including the requirement for substantial computational resources, the complexity of simulating real-world settings and the challenge of guaranteeing the stability and resilience of RL algorithms in dynamic and unpredictable environments. RL has demonstrated significant promise, but more research is needed to fully integrate it into industrial and environmental systems in order to solve the current challenges. Abbreviations: AC: Actor critic; AI: Artificial intelligence; ANN: Artificial neural networks; A3C: Asynchronous advantage actor critic; CRL : Classical Reinforcement learning; CV : Controlled variable; DDPG : Deep deterministic policy gradient; DQN: Deep Q network; DRL: Deep reinforcement learning; DP: Dynamic programming; FOMDP: Fully observable Markov decision process; GRU: Gated recurrent unit; LQR: Linear quadratic regulator; LSTM: Long short-term memory; ML: Machine learning; MV : Manipulated variable; MC: Monte Carlo; MDP: Markov decision process; MPC: Model predictive controller; MIMO: Multi input multi output; PG: Policy gradient; PID: Proportional integral derivative; PPO: Proximal policy optimisation; RL: Reinforcement learning; PPO: Proximal policy optimisation; SAC: Soft actor critic; SISO: Single input single output; TD: Temporal difference; TRPO: Trust region policy optimisation; TD3: Twin delayed deep deterministic policy gradient.

Finite-sample analysis of nonlinear stochastic approximation with applications in reinforcement learning

Reinforcement Learning (RL) algorithms work well with well-defined rewards, but they fail with sparse/deceptive rewards and require additional exploration strategies. This work introduces a deep exploration method based on the Upper Confidence Bound (UCB) bonus. The proposed method can be plugged into actor-critic algorithms that use deep neural networks as a critic. Based on the conclusion of the regret bound under the linear Markov decision process approximation, we use the feature matrix to calculate the UCB bonus for deep exploration. The proposed method is equivalent to the count-based exploration method in special cases and is suitable for general situations. Our method uses the last d-dimensional feature vector in the critic network and is easy to deploy. We design a simple task, “swim”, to demonstrate the principle of the proposed method to achieve exploration in sparse/deceptive reward environments. Then, we perform an empirical evaluation on sparse/deceptive reward version gym environments and Ackermann robot control tasks. The evaluation results verify that the proposed algorithm can perform effective deep explorations in sparse/deceptive reward tasks.

This paper presents an approach to combinatorial optimization problems using linear function approximation (LFA) to solve the Travelling Salesman Problem (TSP). We create a reinforcement learning model in which we parameterize our policy using linear function approximation instead of the more commonly used neural networks. We then evaluated our models based on two factors: training time and optimality. When we compared our results with a state-of-the-art neural network solver, we found that our model was able to solve the TSP accurately while using drastically less computational resources and time to train than the neural network algorithm (Kool et al., 2019).

Advanced planning for autonomous vehicles using reinforcement learning and deep inverse reinforcement learning

We consider primal-dual-based reinforcement learning (RL) in episodic constrained Markov decision processes (CMDPs) with non-stationary objectives and constraints, which plays a central role in ensuring the safety of RL in time-varying environments. In this problem, the reward/utility functions and the state transition functions are both allowed to vary arbitrarily over time as long as their cumulative variations do not exceed certain known variation budgets. Designing safe RL algorithms in time-varying environments is particularly challenging because of the need to integrate the constraint violation reduction, safe exploration, and adaptation to the non-stationarity. To this end, we identify two alternative conditions on the time-varying constraints under which we can guarantee the safety in the long run. We also propose the Periodically Restarted Optimistic Primal-Dual Proximal Policy Optimization (PROPD-PPO) algorithm that can coordinate with both two conditions. Furthermore, a dynamic regret bound and a constraint violation bound are established for the proposed algorithm in both the linear kernel CMDP function approximation setting and the tabular CMDP setting under two alternative conditions. This paper provides the first provably efficient algorithm for non-stationary CMDPs with safe exploration.

We study the sample complexity of offline learning for a class of structured Markov decision processes (MDPs) describing the inventory control system with fixed ordering cost/setup cost, a fundamental problem in supply chains. We find that a naive plug-in sampling-based approach applied to the inventory MDPs leads to strictly lower sample complexity bounds compared to the optimal bounds recently obtained for the general MDPs. More specifically, in the infinite-horizon discounted cost setting, we obtain an O ~ ( min { ( S ¯ − s _ ) 2 ( 1 − γ ) 2 ϵ 2 , 1 ( 1 − γ ) 4 ϵ 2 } ) sample complexity bound, where ( S ¯ − s _ ) 2 corresponds to the number of state-action pairs in a generic MDP with state space S and action space A . As such, O ~ ( ( S ¯ − s _ ) 2 ( 1 − γ ) 2 ϵ 2 ) improves on the optimal generic reinforcement learning (RL) bound Θ ~ ( ( S ¯ − s _ ) 2 ( 1 − γ ) 3 ϵ 2 ) (when directly applying Θ ~ ( | S | | A | ( 1 − γ ) 3 ϵ 2 ) here) by a factor of ( 1 − γ ) − 1 , and O ~ ( 1 ( 1 − γ ) 4 ϵ 2 ) is able to completely remove the dependence on state and action cardinality. In the infinite-horizon average cost setting, we obtain an O ~ ( ( S ¯ − s _ ) 2 ϵ 2 ) bound, improving on the generic optimal RL bound Θ ~ ( ( S ¯ − s _ ) 2 t m i x ϵ 2 ) (when directly applying Θ ~ ( | S | | A | t m i x ϵ 2 ) here) by a factor of t m i x , and hence removing the mixing time dependence. By carefully leveraging the structural properties of the inventory dynamics in various settings, we are able to improve on those “best-possible” bounds developed in the RL literature. Our results demonstrate the drawbacks one could face by blindly following RL algorithms and the necessity of designing sample efficient algorithms that properly incorporate the special structures of the inventory systems.

Reinforcement learning (RL) in problems with large markov decision processes (MDPs) can result in many action policies. Over time, it becomes non-trivial computationally to search for action policies that are effective to the states. Function approximation can be used to generalize the action policies to improve the search efficiency. Based on the Adaptive Resonance Theory, FASON is a type of self-organizing neural networks that can be used for function approximation. This paper introduces the features that allow FASON to act and learn efficiently in large MDPs. Being structurally compatible to the propositional rules, the performance of FASON can be improved by directly inserting not-easily-learned domain knowledge into it. For speeding up RL, the Knowledge-based Exploration strategy is used to direct exploration. To select cognitive nodes effective to the states, a Reward Vigilance Adaptation strategy is used to adapt the reward vigilance. The performance of FASON is compared with selected benchmark models in three RL problems with large MDPs. Ten test cases were designed to study the effect of four key parameters on FASON. Experiment results of the winning parameters are presented to inform on the best ways of using FASON.