On the Divergence of Differential Temporal Difference Learning without Local Clocks

Abstract

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 $\alpha_t$ that depends only on the time step $t$ (i.e., a global clock). The latter is of the form $\alpha_{\nu(S_t, t)}$, where $\nu(s, t)$ 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 (TD) learning is convergent with a local clock, it can diverge with a global clock. We empirically observe this divergence in our experiments and also prove that the associated ODE of differential TD with a global clock is unstable. This counterexample closes the open problem in Wan et al. [2021], Blaser et al. [2026].