Model approximation in MDPs with unbounded per-step cost

We consider the problem of designing a control policy for an infinite-horizon discounted cost Markov decision process $\mathcal{M}$ when we only have access to an approximate model $\hat{\mathcal{M}}$. How well does an optimal policy $\hat{\pi}^{\star}$ of the approximate model perform when used in the original model $\mathcal{M}$? We answer this question by bounding a weighted norm of the difference between the value function of $\hat{\pi}^\star $ when used in $\mathcal{M}$ and the optimal value function of $\mathcal{M}$. We then extend our results and obtain potentially tighter upper bounds by considering affine transformations of the per-step cost. We further provide upper bounds that explicitly depend on the weighted distance between cost functions and weighted distance between transition kernels of the original and approximate models. We present examples to illustrate our results.

Bridging State and History Representations: Understanding Self-Predictive RL

Representations are at the core of all deep reinforcement learning (RL) methods for both Markov decision processes (MDPs) and partially observable Markov decision processes (POMDPs). Many representation learning methods and theoretical frameworks have been developed to understand what constitutes an effective representation. However, the relationships between these methods and the shared properties among them remain unclear. In this paper, we show that many of these seemingly distinct methods and frameworks for state and history abstractions are, in fact, based on a common idea of self-predictive abstraction. Furthermore, we provide theoretical insights into the widely adopted objectives and optimization, such as the stop-gradient technique, in learning self-predictive representations. These findings together yield a minimalist algorithm to learn self-predictive representations for states and histories. We validate our theories by applying our algorithm to standard MDPs, MDPs with distractors, and POMDPs with sparse rewards. These findings culminate in a set of practical guidelines for RL practitioners.

Approximate information state based convergence analysis of recurrent Q-learning

In spite of the large literature on reinforcement learning (RL) algorithms for partially observable Markov decision processes (POMDPs), a complete theoretical understanding is still lacking. In a partially observable setting, the history of data available to the agent increases over time so most practical algorithms either truncate the history to a finite window or compress it using a recurrent neural network leading to an agent state that is non-Markovian. In this paper, it is shown that in spite of the lack of the Markov property, recurrent Q-learning (RQL) converges in the tabular setting. Moreover, it is shown that the quality of the converged limit depends on the quality of the representation which is quantified in terms of what is known as an approximate information state (AIS). Based on this characterization of the approximation error, a variant of RQL with AIS losses is presented. This variant performs better than a strong baseline for RQL that does not use AIS losses. It is demonstrated that there is a strong correlation between the performance of RQL over time and the loss associated with the AIS representation.

Dealing With Non-stationarity in Decentralized Cooperative Multi-Agent Deep Reinforcement Learning via Multi-Timescale Learning

Decentralized cooperative multi-agent deep reinforcement learning (MARL) can be a versatile learning framework, particularly in scenarios where centralized training is either not possible or not practical. One of the key challenges in decentralized deep MARL is the non-stationarity of the learning environment when multiple agents are learning concurrently. A commonly used and efficient scheme for decentralized MARL is independent learning in which agents concurrently update their policies independent of each other. We first show that independent learning does not always converge, while sequential learning where agents update their policies one after another in a sequence is guaranteed to converge to an agent-by-agent optimal solution. In sequential learning, when one agent updates its policy, all other agent's policies are kept fixed, alleviating the challenge of non-stationarity due to concurrent updates in other agents' policies. However, it can be slow because only one agent is learning at any time. Therefore it might also not always be practical. In this work, we propose a decentralized cooperative MARL algorithm based on multi-timescale learning. In multi-timescale learning, all agents learn concurrently, but at different learning rates. In our proposed method, when one agent updates its policy, other agents are allowed to update their policies as well, but at a slower rate. This speeds up sequential learning, while also minimizing non-stationarity caused by other agents updating concurrently. Multi-timescale learning outperforms state-of-the-art decentralized learning methods on a set of challenging multi-agent cooperative tasks in the epymarl (papoudakis2020) benchmark. This can be seen as a first step towards more general decentralized cooperative deep MARL methods based on multi-timescale learning.

On learning history based policies for controlling Markov decision processes

Reinforcementlearning(RL)folkloresuggeststhathistory-basedfunctionapproximationmethods,suchas recurrent neural nets or history-based state abstraction, perform better than their memory-less counterparts, due to the fact that function approximation in Markov decision processes (MDP) can be viewed as inducing a Partially observable MDP. However, there has been little formal analysis of such history-based algorithms, as most existing frameworks focus exclusively on memory-less features. In this paper, we introduce a theoretical framework for studying the behaviour of RL algorithms that learn to control an MDP using history-based feature abstraction mappings. Furthermore, we use this framework to design a practical RL algorithm and we numerically evaluate its effectiveness on a set of continuous control tasks.

On learning Whittle index policy for restless bandits with scalable regret

Reinforcement learning is an attractive approach to learn good resource allocation and scheduling policies based on data when the system model is unknown. However, the cumulative regret of most RL algorithms scales as $\tilde O(\mathsf{S} \sqrt{\mathsf{A} T})$, where $\mathsf{S}$ is the size of the state space, $\mathsf{A}$ is the size of the action space, $T$ is the horizon, and the $\tilde{O}(\cdot)$ notation hides logarithmic terms. Due to the linear dependence on the size of the state space, these regret bounds are prohibitively large for resource allocation and scheduling problems. In this paper, we present a model-based RL algorithm for such problem which has scalable regret. In particular, we consider a restless bandit model, and propose a Thompson-sampling based learning algorithm which is tuned to the underlying structure of the model. We present two characterizations of the regret of the proposed algorithm with respect to the Whittle index policy. First, we show that for a restless bandit with $n$ arms and at most $m$ activations at each time, the regret scales either as $\tilde{O}(mn\sqrt{T})$ or $\tilde{O}(n^2 \sqrt{T})$ depending on the reward model. Second, under an additional technical assumption, we show that the regret scales as $\tilde{O}(n^{1.5} \sqrt{T})$. We present numerical examples to illustrate the salient features of the algorithm.

Consistency and Rate of Convergence of Switched Least Squares System Identification for Autonomous Switched Linear Systems

In this paper, we investigate the problem of system identification for autonomous switched linear systems with complete state observations. We propose switched least squares method for the identification for switched linear systems, show that this method is strongly consistent, and derive data-dependent and data-independent rates of convergence. In particular, our data-dependent rate of convergence shows that, almost surely, the system identification error is $\mathcal{O}\big(\sqrt{\log(T)/T} \big)$ where $T$ is the time horizon. These results show that our method for switched linear systems has the same rate of convergence as least squares method for non-switched linear systems. We compare our results with those in the literature. We present numerical examples to illustrate the performance of the proposed system identification method.

Scalable Operator Allocation for Multi-Robot Assistance: A Restless Bandit Approach

In this paper, we consider the problem of allocating human operators in a system with multiple semi-autonomous robots. Each robot is required to perform an independent sequence of tasks, subjected to a chance of failing and getting stuck in a fault state at every task. If and when required, a human operator can assist or teleoperate a robot. Conventional MDP techniques used to solve such problems face scalability issues due to exponential growth of state and action spaces with the number of robots and operators. In this paper we derive conditions under which the operator allocation problem is indexable, enabling the use of the Whittle index heuristic. The conditions can be easily checked to verify indexability, and we show that they hold for a wide range of problems of interest. Our key insight is to leverage the structure of the value function of individual robots, resulting in conditions that can be verified separately for each state of each robot. We apply these conditions to two types of transitions commonly seen in remote robot supervision systems. Through numerical simulations, we demonstrate the efficacy of Whittle index policy as a near-optimal and scalable approach that outperforms existing scalable methods.

A relaxed technical assumption for posterior sampling-based reinforcement learning for control of unknown linear systems

We revisit the Thompson sampling algorithm to control an unknown linear quadratic (LQ) system recently proposed by Ouyang et al (arXiv:1709.04047). The regret bound of the algorithm was derived under a technical assumption on the induced norm of the closed loop system. In this technical note, we show that by making a minor modification in the algorithm (in particular, ensuring that an episode does not end too soon), this technical assumption on the induced norm can be replaced by a milder assumption in terms of the spectral radius of the closed loop system. The modified algorithm has the same Bayesian regret of $\tilde{\mathcal{O}}(\sqrt{T})$, where $T$ is the time-horizon and the $\tilde{\mathcal{O}}(\cdot)$ notation hides logarithmic terms in~$T$.

Scalable regret for learning to control network-coupled subsystems with unknown dynamics

We consider the problem of controlling an unknown linear quadratic Gaussian (LQG) system consisting of multiple subsystems connected over a network. Our goal is to minimize and quantify the regret (i.e. loss in performance) of our strategy with respect to an oracle who knows the system model. Viewing the interconnected subsystems globally and directly using existing LQG learning algorithms for the global system results in a regret that increases super-linearly with the number of subsystems. Instead, we propose a new Thompson sampling based learning algorithm which exploits the structure of the underlying network. We show that the expected regret of the proposed algorithm is bounded by $\tilde{\mathcal{O}} \big( n \sqrt{T} \big)$ where $n$ is the number of subsystems, $T$ is the time horizon and the $\tilde{\mathcal{O}}(\cdot)$ notation hides logarithmic terms in $n$ and $T$. Thus, the regret scales linearly with the number of subsystems. We present numerical experiments to illustrate the salient features of the proposed algorithm.