Approximate Value Equivalence

Model-based reinforcement learning agents must make compromises about which aspects of the environment their models should capture. The value equivalence (VE) principle posits that these compromises should be made considering the model's eventual use in value-based planning. Given sets of functions and policies, a model is said to be order-$k$ VE to the environment if $k$ applications of the Bellman operators induced by the policies produce the correct result when applied to the functions. Prior work investigated the classes of models induced by VE when we vary $k$ and the sets of policies and functions. This gives rise to a rich collection of topological relationships and conditions under which VE models are optimal for planning. Despite this effort, relatively little is known about the planning performance of models that fail to satisfy these conditions. This is due to the rigidity of the VE formalism, as classes of VE models are defined with respect to \textit{exact} constraints on their Bellman operators. This limitation gets amplified by the fact that such constraints themselves may depend on functions that can only be approximated in practice. To address these problems we propose approximate value equivalence (AVE), which extends the VE formalism by replacing equalities with error tolerances. This extension allows us to show that AVE models with respect to one set of functions are also AVE with respect to any other set of functions if we tolerate a high enough error. We can then derive bounds on the performance of VE models with respect to \textit{arbitrary sets of functions}. Moreover, AVE models more accurately reflect what can be learned by our agents in practice, allowing us to investigate previously unexplored tensions between model capacity and the choice of VE model class. In contrast to previous works, we show empirically that there are situations where agents with limited capacity should prefer to learn more accurate models with respect to smaller sets of functions over less accurate models with respect to larger sets of functions.