A Definition of Continual Reinforcement Learning

In this paper we develop a foundation for continual reinforcement learning.

On the Convergence of Bounded Agents

When has an agent converged? Standard models of the reinforcement learning problem give rise to a straightforward definition of convergence: An agent converges when its behavior or performance in each environment state stops changing. However, as we shift the focus of our learning problem from the environment's state to the agent's state, the concept of an agent's convergence becomes significantly less clear. In this paper, we propose two complementary accounts of agent convergence in a framing of the reinforcement learning problem that centers around bounded agents. The first view says that a bounded agent has converged when the minimal number of states needed to describe the agent's future behavior cannot decrease. The second view says that a bounded agent has converged just when the agent's performance only changes if the agent's internal state changes. We establish basic properties of these two definitions, show that they accommodate typical views of convergence in standard settings, and prove several facts about their nature and relationship. We take these perspectives, definitions, and analysis to bring clarity to a central idea of the field.

Settling the Reward Hypothesis

The reward hypothesis posits that, "all of what we mean by goals and purposes can be well thought of as maximization of the expected value of the cumulative sum of a received scalar signal (reward)." We aim to fully settle this hypothesis. This will not conclude with a simple affirmation or refutation, but rather specify completely the implicit requirements on goals and purposes under which the hypothesis holds.

Meta-Gradients in Non-Stationary Environments

Meta-gradient methods (Xu et al., 2018; Zahavy et al., 2020) offer a promising solution to the problem of hyperparameter selection and adaptation in non-stationary reinforcement learning problems. However, the properties of meta-gradients in such environments have not been systematically studied. In this work, we bring new clarity to meta-gradients in non-stationary environments. Concretely, we ask: (i) how much information should be given to the learned optimizers, so as to enable faster adaptation and generalization over a lifetime, (ii) what meta-optimizer functions are learned in this process, and (iii) whether meta-gradient methods provide a bigger advantage in highly non-stationary environments. To study the effect of information provided to the meta-optimizer, as in recent works (Flennerhag et al., 2021; Almeida et al., 2021), we replace the tuned meta-parameters of fixed update rules with learned meta-parameter functions of selected context features. The context features carry information about agent performance and changes in the environment and hence can inform learned meta-parameter schedules. We find that adding more contextual information is generally beneficial, leading to faster adaptation of meta-parameter values and increased performance over a lifetime. We support these results with a qualitative analysis of resulting meta-parameter schedules and learned functions of context features. Lastly, we find that without context, meta-gradients do not provide a consistent advantage over the baseline in highly non-stationary environments. Our findings suggest that contextualizing meta-gradients can play a pivotal role in extracting high performance from meta-gradients in non-stationary settings.

A Theory of Abstraction in Reinforcement Learning

Reinforcement learning defines the problem facing agents that learn to make good decisions through action and observation alone. To be effective problem solvers, such agents must efficiently explore vast worlds, assign credit from delayed feedback, and generalize to new experiences, all while making use of limited data, computational resources, and perceptual bandwidth. Abstraction is essential to all of these endeavors. Through abstraction, agents can form concise models of their environment that support the many practices required of a rational, adaptive decision maker. In this dissertation, I present a theory of abstraction in reinforcement learning. I first offer three desiderata for functions that carry out the process of abstraction: they should 1) preserve representation of near-optimal behavior, 2) be learned and constructed efficiently, and 3) lower planning or learning time. I then present a suite of new algorithms and analysis that clarify how agents can learn to abstract according to these desiderata. Collectively, these results provide a partial path toward the discovery and use of abstraction that minimizes the complexity of effective reinforcement learning.

On the Expressivity of Markov Reward

Reward is the driving force for reinforcement-learning agents. This paper is dedicated to understanding the expressivity of reward as a way to capture tasks that we would want an agent to perform. We frame this study around three new abstract notions of "task" that might be desirable: (1) a set of acceptable behaviors, (2) a partial ordering over behaviors, or (3) a partial ordering over trajectories. Our main results prove that while reward can express many of these tasks, there exist instances of each task type that no Markov reward function can capture. We then provide a set of polynomial-time algorithms that construct a Markov reward function that allows an agent to optimize tasks of each of these three types, and correctly determine when no such reward function exists. We conclude with an empirical study that corroborates and illustrates our theoretical findings.

Bad-Policy Density: A Measure of Reinforcement Learning Hardness

Reinforcement learning is hard in general. Yet, in many specific environments, learning is easy. What makes learning easy in one environment, but difficult in another? We address this question by proposing a simple measure of reinforcement-learning hardness called the bad-policy density. This quantity measures the fraction of the deterministic stationary policy space that is below a desired threshold in value. We prove that this simple quantity has many properties one would expect of a measure of learning hardness. Further, we prove it is NP-hard to compute the measure in general, but there are paths to polynomial-time approximation. We conclude by summarizing potential directions and uses for this measure.

Control of mental representations in human planning

One of the most striking features of human cognition is the capacity to plan. Two aspects of human planning stand out: its efficiency, even in complex environments, and its flexibility, even in changing environments. Efficiency is especially impressive because directly computing an optimal plan is intractable, even for modestly complex tasks, and yet people successfully solve myriad everyday problems despite limited cognitive resources. Standard accounts in psychology, economics, and artificial intelligence have suggested this is because people have a mental representation of a task and then use heuristics to plan in that representation. However, this approach generally assumes that mental representations are fixed. Here, we propose that mental representations can be controlled and that this provides opportunities to adaptively simplify problems so they can be more easily reasoned about -- a process we refer to as construal. We construct a formal model of this process and, in a series of large, pre-registered behavioral experiments, show both that construal is subject to online cognitive control and that people form value-guided construals that optimally balance the complexity of a representation and its utility for planning and acting. These results demonstrate how strategically perceiving and conceiving problems facilitates the effective use of limited cognitive resources.

Revisiting Peng's Q($λ$) for Modern Reinforcement Learning

Off-policy multi-step reinforcement learning algorithms consist of conservative and non-conservative algorithms: the former actively cut traces, whereas the latter do not. Recently, Munos et al. (2016) proved the convergence of conservative algorithms to an optimal Q-function. In contrast, non-conservative algorithms are thought to be unsafe and have a limited or no theoretical guarantee. Nonetheless, recent studies have shown that non-conservative algorithms empirically outperform conservative ones. Motivated by the empirical results and the lack of theory, we carry out theoretical analyses of Peng's Q($\lambda$), a representative example of non-conservative algorithms. We prove that it also converges to an optimal policy provided that the behavior policy slowly tracks a greedy policy in a way similar to conservative policy iteration. Such a result has been conjectured to be true but has not been proven. We also experiment with Peng's Q($\lambda$) in complex continuous control tasks, confirming that Peng's Q($\lambda$) often outperforms conservative algorithms despite its simplicity. These results indicate that Peng's Q($\lambda$), which was thought to be unsafe, is a theoretically-sound and practically effective algorithm.

What can I do here? A Theory of Affordances in Reinforcement Learning

Reinforcement learning algorithms usually assume that all actions are always available to an agent. However, both people and animals understand the general link between the features of their environment and the actions that are feasible. Gibson (1977) coined the term "affordances" to describe the fact that certain states enable an agent to do certain actions, in the context of embodied agents. In this paper, we develop a theory of affordances for agents who learn and plan in Markov Decision Processes. Affordances play a dual role in this case. On one hand, they allow faster planning, by reducing the number of actions available in any given situation. On the other hand, they facilitate more efficient and precise learning of transition models from data, especially when such models require function approximation. We establish these properties through theoretical results as well as illustrative examples. We also propose an approach to learn affordances and use it to estimate transition models that are simpler and generalize better.