On the Divergence of Differential Temporal Difference Learning without Local Clocks

Learning rate is a critical component of reinforcement learning (RL). This work uses global and local clocks to distinguish two types of learning rates. The former is of the standard form $α_t$ that depends only on the time step $t$ (i.e., a global clock). The latter is of the form $α_{ν(S_t, t)}$, where $ν(s, t)$ counts the number of visits to state $s$ until time $t$ (i.e., a local clock). In discounted RL, an RL algorithm that is convergent with a local clock is always also convergent with a global clock, and vice versa. We are not aware of any counterexample. The key contribution of this work is to show that this nice correspondence breaks down in average-reward RL. Specifically, we construct a counterexample showing that although differential temporal difference learning is convergent with a local clock, it can diverge with a global clock. This counterexample closes the open problem in Wan et al. [2021], Blaser et al. [2026].