Implementation Filters and Delay-Budget Instability in Coupled Replicator--Mutator Dynamics

What the math gives to ML

The paper isolates a useful distinction in delayed feedback systems: observation and deployment delays contribute only through one additive phase budget, while first-order implementation filters add independent poles that can destabilize the loop even when hard delay is zero. This structure transfers naturally to coupled neural training systems such as GANs, actor-critics, minimax optimization, and asynchronous two-model training, where gradients or opponent parameters are stale and EMA or optimizer states act as implementation filters. The practical use is a delay-budget controller that estimates the dominant coupled-mode gain and distinguishes recoverable delay-induced oscillation from filter-induced instability. A small-signal Hopf test can provide an actionable warning and guide staleness, EMA, or relaxation schedules.