Why Should I Trust You, Bellman? The Bellman Error is a Poor Replacement for Value Error
This work addresses a foundational issue in reinforcement learning for researchers and practitioners, revealing limitations in a widely used method, though it is incremental in analyzing existing concepts.
The paper tackles the problem of using the Bellman error as a surrogate objective for value prediction accuracy in reinforcement learning, showing that it is a poor proxy because its magnitude is weakly related to the distance to the true value function and it can be minimized without improving accuracy.
In this work, we study the use of the Bellman equation as a surrogate objective for value prediction accuracy. While the Bellman equation is uniquely solved by the true value function over all state-action pairs, we find that the Bellman error (the difference between both sides of the equation) is a poor proxy for the accuracy of the value function. In particular, we show that (1) due to cancellations from both sides of the Bellman equation, the magnitude of the Bellman error is only weakly related to the distance to the true value function, even when considering all state-action pairs, and (2) in the finite data regime, the Bellman equation can be satisfied exactly by infinitely many suboptimal solutions. This means that the Bellman error can be minimized without improving the accuracy of the value function. We demonstrate these phenomena through a series of propositions, illustrative toy examples, and empirical analysis in standard benchmark domains.