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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. 

We present Q-functionals, an alternative architecture for continuous control deep reinforcement learning. Instead of returning a single value for a state-action pair, our network transforms a state into a function that can be rapidly evaluated in parallel for many actions, allowing us to efficiently choose high-value actions through sampling. This contrasts with the typical architecture of off-policy continuous control, where a policy network is trained for the sole purpose of selecting actions from the Q-function. We represent our action-dependent Q-function as a weighted sum of basis functions (Fourier, Polynomial, etc) over the action space, where the weights are state-dependent and output by the Q-functional network. Fast sampling makes practical a variety of techniques that require Monte-Carlo integration over Q-functions, and enables action-selection strategies besides simple value-maximization. We characterize our framework, describe various implementations of Q-functionals, and demonstrate strong performance on a suite of continuous control tasks. 

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.