We consider how to construct state abstractions compatible with a given set of abstract actions, to obtain a well-formed abstract Markov decision process (MDP). We show that the Bellman equation suggests that abstract states should represent distributions over states in the ground MDP; we characterize the conditions under which the resulting process is Markov and approximately model-preserving, derive algorithms for constructing and planning with the abstract MDP, and apply them to a visual maze task. We generalize these results to the factored actions case, characterizing the conditions that result in factored abstract states and apply the resulting algorithm to Montezuma's Revenge. These results provide a powerful and principled framework for constructing neurosymbolic abstract Markov decision processes.

Reinforcement learning algorithms typically rely on the assumption that the environment dynamics and value function can be expressed in terms of a Markovian state representation. However, when state information is only partially observable, how can an agent learn such a state representation, and how can it detect when it has found one? We introduce a metric that can accomplish both objectives, without requiring access to---or knowledge of---an underlying, unobservable state space. Our metric, the λ-discrepancy, is the difference between two distinct temporal difference (TD) value estimates, each computed using TD(λ) with a different value of λ. Since TD(λ=0) makes an implicit Markov assumption and TD(λ=1) does not, a discrepancy between these estimates is a potential indicator of a non-Markovian state representation. Indeed, we prove that the λ-discrepancy is exactly zero for all Markov decision processes and almost always non-zero for a broad class of partially observable environments. We also demonstrate empirically that, once detected, minimizing the λ-discrepancy can help with learning a memory function to mitigate the corresponding partial observability. We then train a reinforcement learning agent that simultaneously constructs two recurrent value networks with different λ parameters and minimizes the difference between them as an auxiliary loss. The approach scales to challenging partially observable domains, where the resulting agent frequently performs significantly better (and never performs worse) than a baseline recurrent agent with only a single value network.

Do neural networks learn to implement algorithms such as look-ahead or search "in the wild"? Or do they rely purely on collections of simple heuristics? We present evidence of *learned look-ahead* in the policy and value network of Leela Chess Zero, the currently strongest deep neural chess engine. We find that Leela internally represents future optimal moves and that these representations are crucial for its final output in certain board states. Concretely, we exploit the fact that Leela is a transformer that treats every chessboard square like a token in language models, and give three lines of evidence: (1) activations on certain squares of future moves are unusually important causally; (2) we find attention heads that move important information "forward and backward in time," e.g., from squares of future moves to squares of earlier ones; and (3) we train a simple probe that can predict the optimal move 2 turns ahead with 92% accuracy (in board states where Leela finds a single best line). These findings are clear evidence of learned look-ahead in neural networks and might be a step towards a better understanding of their capabilities.

Principled decision-making in continuous state–action spaces is impossible without some assumptions. A common approach is to assume Lipschitz continuity of the Q-function. We show that, unfortunately, this property fails to hold in many typical domains. We propose a new coarse-grained smoothness definition that generalizes the notion of Lipschitz continuity, is more widely applicable, and allows us to compute significantly tighter bounds on Q-functions, leading to improved learning. We provide a theoretical analysis of our new smoothness definition, and discuss its implications and impact on control and exploration in continuous domains.

Optimistic initialization underpins many theoretically sound exploration schemes in tabular domains; however, in the deep function approximation setting, optimism can quickly disappear if initialized naively. We propose a framework for more effectively incorporating optimistic initialization into reinforcement learning for continuous control. Our approach uses metric information about the state-action space to estimate which transitions are still unexplored, and explicitly maintains the initial Q-value optimism for the corresponding state-action pairs. We also develop methods for efficiently approximating these training objectives, and for incorporating domain knowledge into the optimistic envelope to improve sample efficiency. We empirically evaluate these approaches on a variety of hard exploration problems in continuous control, where our method outperforms existing exploration techniques.

The difficulty of deterministic planning increases exponentially with search-tree depth. Black-box planning presents an even greater challenge, since planners must operate without an explicit model of the domain. Heuristics can make search more efficient, but goal-aware heuristics for black-box planning usually rely on goal counting, which is often quite uninformative. In this work, we show how to overcome this limitation by discovering macro-actions that make the goal-count heuristic more accurate. Our approach searches for macro-actions with focused effects (i.e. macros that modify only a small number of state variables), which align well with the assumptions made by the goal-count heuristic. Focused macros dramatically improve black-box planning efficiency across a wide range of planning domains, sometimes beating even state-of-the-art planners with access to a full domain model.

A fundamental assumption of reinforcement learning in Markov decision processes (MDPs) is that the relevant decision process is, in fact, Markov. However, when MDPs have rich observations, agents typically learn by way of an abstract state representation, and such representations are not guaranteed to preserve the Markov property. We introduce a novel set of conditions and prove that they are sufficient for learning a Markov abstract state representation. We then describe a practical training procedure that combines inverse model estimation and temporal contrastive learning to learn an abstraction that approximately satisfies these conditions. Our novel training objective is compatible with both online and offline training: it does not require a reward signal, but agents can capitalize on reward information when available. We empirically evaluate our approach on a visual gridworld domain and a set of continuous control benchmarks. Our approach learns representations that capture the underlying structure of the domain and lead to improved sample efficiency over state-of-the-art deep reinforcement learning with visual features---often matching or exceeding the performance achieved with hand-designed compact state information.