FOAM: Frequency and Operator Error-Based Adaptive Damping Method for Reducing Staleness-Oriented Error for Shampoo

ArXi:2606.02365v1 Announce Type: cross Shampoo is attracting considerable attention for its superior performance on large-scale optimization benchmarks; yet it faces a significant practical bottleneck: the prohibitive computational overhead of matrix inversion. To mitigate this, practitioners typically rely on stale preconditioner updates, creating a fundamental trade-off between computational efficiency and optimization fidelity. In this work, we provide a theoretical study of staleness through the complementary lenses of convergence and stability.