  Recent progress has been made in understanding optimisation dynamics in
neural networks trained with full-batch gradient descent with momentum with the
uncovering of the edge of stability phenomenon in supervised learning. The edge
of stability phenomenon occurs as the leading eigenvalue of the Hessian reaches
the divergence threshold of the underlying optimisation algorithm for a
quadratic loss, after which it starts oscillating around the threshold, and the
loss starts to exhibit local instability but decreases over long time frames.
In this work, we explore the edge of stability phenomenon in reinforcement
learning (RL), specifically off-policy Q-learning algorithms across a variety
of data regimes, from offline to online RL. Our experiments reveal that,
despite significant differences to supervised learning, such as
non-stationarity of the data distribution and the use of bootstrapping, the
edge of stability phenomenon can be present in off-policy deep RL. Unlike
supervised learning, however, we observe strong differences depending on the
underlying loss, with DQN -- using a Huber loss -- showing a strong edge of
stability effect that we do not observe with C51 -- using a cross entropy loss.
Our results suggest that, while neural network structure can lead to
optimisation dynamics that transfer between problem domains, certain aspects of
deep RL optimisation can differentiate it from domains such as supervised
learning.

Machine Learning