Why Smooth Stability Assumptions Fail for ReLU Learning

Abstract

Stability analyses of modern learning systems are frequently derived under smoothness assumptions that are violated by ReLU-type nonlinearities. In this note, we isolate a minimal obstruction by showing that no uniform smoothness-based stability proxy such as gradient Lipschitzness or Hessian control can hold globally for ReLU networks, even in simple settings where training trajectories appear empirically stable. We give a concrete counterexample demonstrating the failure of classical stability bounds and identify a minimal generalized derivative condition under which stability statements can be meaningfully restored. The result clarifies why smooth approximations of ReLU can be misleading and motivates nonsmooth-aware stability frameworks.