Weber–Fechner law in temporal difference learning derived from control as inference

This study investigates a novel nonlinear update rule for value and policy functions based on temporal difference (TD) errors in reinforcement learning (RL). The update rule in standard RL states that the TD error is linearly proportional to the degree of updates, treating all rewards equally without any bias. On the other hand, recent biological studies have revealed that there are nonlinearities in the TD error and the degree of updates, biasing policies towards being either optimistic or pessimistic. Such biases in learning due to nonlinearities are expected to be useful and intentionally leftover features in biological learning. Therefore, this research explores a theoretical framework that can leverage the nonlinearity between the degree of the update and TD errors. To this end, we focus on a <i>control as inference</i> framework utilized in the previous work, in which the uncomputable nonlinear term needed to be approximately excluded from the derivation of the standard RL. By analyzing it, the Weber-Fechner law (WFL) is found, in which perception (i.e., the degree of updates) in response to a change in stimulus (i.e., TD error) is attenuated as the stimulus intensity (i.e., the value function) increases. To numerically demonstrate the utilities of WFL on RL, we propose a practical implementation using a reward-punishment framework and modify the definition of optimality. Further analysis of this implementation reveals that two utilities can be expected: i) to accelerate escaping from the situations with small rewards and ii) to pursue the minimum punishment as much as possible. We finally investigate and discuss the expected utilities through simulations and robot experiments. As a result, the proposed RL algorithm with WFL shows the expected utilities that accelerate the reward-maximizing startup and continue to suppress punishments during learning.