Why Smooth Stability Assumptions Fail for ReLU Learning

TL;DR

This paper demonstrates that classical smoothness-based stability analyses fail for ReLU neural networks, providing counterexamples and proposing a generalized derivative condition to better understand stability in nonsmooth settings.

Contribution

It identifies the fundamental limitations of smoothness assumptions in ReLU networks and introduces a minimal generalized derivative condition to restore meaningful stability analysis.

Findings

Classical stability bounds do not hold for ReLU networks.

Counterexamples show failure of gradient Lipschitzness in simple settings.

A generalized derivative condition can restore stability insights.

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.