Increasing the Action Gap: New Operators for Reinforcement Learning

This paper introduces new optimality-preserving operators on Q-functions. We first describe an operator for tabular representations, the consistent Bellman operator, which incorporates a notion of local policy consistency. We show that this local consistency leads to an increase in the action gap at each state; increasing this gap, we argue, mitigates the undesirable effects of approximation and estimation errors on the induced greedy policies. This operator can also be applied to discretized continuous space and time problems, and we provide empirical results evidencing superior performance in this context. Extending the idea of a locally consistent operator, we then derive sufficient conditions for an operator to preserve optimality, leading to a family of operators which includes our consistent Bellman operator. As corollaries we provide a proof of optimality for Baird’s advantage learning algorithm and derive other gap-increasing operators with interesting properties. We conclude with an empirical study on 60 Atari 2600 games illustrating the strong potential of these new operators. Value-based reinforcement learning is an attractive solution to planning problems in environments with unknown, unstructured dynamics. In its canonical form, value-based reinforcement learning produces successive refinements of an initial value function through repeated application of a convergent operator. In particular, value iteration (Bellman 1957) directly computes the value function through the iterated evaluation of Bellman’s equation, either exactly or from samples (e.g. Q-Learning, Watkins 1989). In its simplest form, value iteration begins with an initial value function V0 and successively computes Vk+1 := T Vk, where T is the Bellman operator. When the environment dynamics are unknown, Vk is typically replaced by Qk, the state-action value function, and T is approximated by an empirical Bellman operator. The fixed point of the Bellman operator, Q∗, is the optimal state-action value function or optimal Q-function, from which an optimal policy π∗ can be recovered. In this paper we argue that the optimal Q-function is inconsistent, in the sense that for any action a which is subop∗Now at Carnegie Mellon University.  While preserving global consistency appears impractical, we propose a simple modification to the Bellman operator which provides us a with a first-order solution to the inconsistency problem. Accordingly, we call our new operator the consistent Bellman operator. We show that the consistent Bellman operator generally devalues suboptimal actions but preserves the set of optimal policies. As a result, the action gap – the value difference between optimal and second best actions – increases. This increasing of the action gap is advantageous in the presence of approximation or estimation error, and may be crucial for systems operating at a fine time scale such as video games (Togelius et al. 2009; Bellemare et al. 2013), real-time markets (Jiang and Powell 2015), and robotic platforms (Riedmiller et al. 2009; Hoburg and Tedrake 2009; Deisenroth and Rasmussen 2011; Sutton et al. 2011). In fact, the idea of devaluating suboptimal actions underpins Baird’s advantage learning (Baird 1999), designed for continuous time control, and occurs naturally when considering the discretized solution of continuous time and space MDPs (e.g. Munos and Moore 1998; 2002), whose limit is the HamiltonJacobi-Bellman equation (Kushner and Dupuis 2001). Our empirical results on the bicycle domain (Randlov and Alstrom 1998) show a marked increase in performance from using the consistent Bellman operator. In the second half of this paper we derive novel sufficient conditions for an operator to preserve optimality. The relative weakness of these new conditions reveal that it is possible to deviate significantly from the Bellman operator without sacrificing optimality: an optimality-preserving operator needs not be contractive, nor even guarantee convergence of the Q-values for suboptimal actions. While numerous alternatives to the Bellman operator have been put forward (e.g. recently Azar et al. 2011; Bertsekas and Yu 2012), we believe our work to be the first to propose such a major departure from the canonical fixed-point condition required from an optimality-preserving operator. As proof of the richness of this new operator family we describe a few practical instantiations with unique properties. We use our operators to obtain state-of-the-art empirical results on the Arcade Learning Environment (Bellemare et al. 2013). We consider the Deep Q-Network (DQN) architecture of Mnih et al. (2015), replacing only its learning rule with one of our operators. Remarkably, this one-line change produces agents that significantly outperform the original DQN. Our work, we believe, demonstrates the potential impact of rethinking the core components of value-based reinforcement learning.