Contextual Decision Processes with Low Bellman Rank are PAC-Learnable (1610.09512v2)

Abstract: This paper studies systematic exploration for reinforcement learning with rich observations and function approximation. We introduce a new model called contextual decision processes, that unifies and generalizes most prior settings. Our first contribution is a complexity measure, the BeLLMan rank, that we show enables tractable learning of near-optimal behavior in these processes and is naturally small for many well-studied reinforcement learning settings. Our second contribution is a new reinforcement learning algorithm that engages in systematic exploration to learn contextual decision processes with low BeLLMan rank. Our algorithm provably learns near-optimal behavior with a number of samples that is polynomial in all relevant parameters but independent of the number of unique observations. The approach uses BeLLMan error minimization with optimistic exploration and provides new insights into efficient exploration for reinforcement learning with function approximation.

• The paper introduces Olive, an optimism-led exploration algorithm that ensures PAC learning in complex contextual decision processes.
• It defines Bellman rank as a key complexity measure, enabling sample-efficient learning for models like LQRs, POMDPs, and PSRs.
• The research provides theoretical guarantees for function approximation in RL, while identifying computational efficiency as a future challenge.

Contextual Decision Processes with Low BeLLMan Rank are PAC-Learnable

The discussed paper explores the domain of reinforcement learning (RL) aimed at creating sample-efficient algorithms that can handle complex observations and state spaces. It introduces a model termed "Contextual Decision Processes" (CDP), a generalization of traditional models like Markov Decision Processes (MDPs) and Partially Observable MDPs (POMDPs). It addresses the vital problem of solving sequential decision-making tasks with rich sensory inputs and advanced function approximation techniques, introducing rigorous theoretical guarantees that were previously lacking for such complex settings.

A principal contribution of the paper is the introduction of the notion of the "BeLLMan rank," a complexity measure that reflects the tractability of learning near-optimal behavior in CDPs. The BeLLMan rank effectively captures the mathematical intertwining between the CDP and the function approximation involved, being naturally small for many classes of problems such as tabular MDPs and low-rank MDPs. Esteemed models including Linear Quadratic Regulators (LQRs), POMDPs, and Predictive State Representations (PSRs) thus fall under contexts that exhibit low BeLLMan ranks. The advantage of employing BeLLMan rank is that it allows these CDPs to be PAC-learnable, i.e., learnable within established bounds of Probably Approximately Correct (PAC) learning.

The paper sets out to resolve challenges by introducing Olive (Optimism Led Iterative Value-function Elimination), a novel algorithm for episodic RL in CDPs. Olive engages in an optimism-driven exploration strategy, enabling it to efficiently handle problems with rich contextual data. A salient feature of Olive is its provision for a polynomial sample complexity warrant, independent of the number of unique observations in the environment, which speaks volumes about its efficacy and adaptability to practical problems.

Another significant insight from this work is the use of BeLLMan error minimization with an optimistic exploration strategy, reshaping how exploration in RL with function approximation is perceived. The algorithm, by ensuring optimistic choices during exploration, directs the agent towards context distributions that help validate validity conditions while maximizing policy value.

The PAC guarantee offered by Olive states that it can learn an $\epsilon$-suboptimal policy with a number of samples polynomial in factors such as the BeLLMan rank, problem horizon, and number of actions. Notably, this complexity does not involve the vast context space, significantly benefiting scenarios with large or even infinite observation spaces.

The theoretical implications of this work point toward a strong understanding of the role of function approximation in RL, particularly in dealing with complex contexts where traditional state labels are inadequate. On a practical note, despite resolving certain challenges with sample efficiency, the computational efficiency remains burdensome. The paper openly leaves future research to focus on adapting Olive into a computationally pragmatic tool.

The broader implications could transcend RL into other areas involving complex temporal decision making under uncertainty, potentially spurring adaptive measures in nonlinear control systems, autonomous systems, and predictive models across various domains. The robustness to changes in function representation underscores the algorithm's adaptability, presenting a fertile ground for incorporating advances in policy optimization.

In conclusion, this research marks a substantial leap in RL theory by addressing systematic exploration with a novel, complexity-appropriate framework that holds potential for further advancements, both in understanding and application. Future explorations could be directed to extend Olive’s efficacy to continuous action spaces and incorporate more computationally harmonious techniques to fully leverage these theoretical underpinnings in practical artificial intelligence systems.