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

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. While staleness improves computational efficiency, it inherently degrades performance and introduces numerical instability. Crucially, we identify that damping, acting as a numerical stabilizer, can effectively suppress these negative effects. Guided by this analysis, we propose FOAM, an adaptive algorithm that stabilizes training by dynamically controlling both the damping factor and the eigendecomposition frequency based on an approximation of the staleness-oriented error. Experimental results demonstrate that FOAM reduces wall-clock time compared to standard Shampoo while maintaining robust convergence.

We presented FOAM, a mathematically grounded framework that resolves the computational bottleneck of preconditioned optimization by decoupling spectral sensing from costly refreshes. By introducing an error proxy and a feedback controller for adaptive damping, our method ensures stability while reducing the frequency of eigendecompositions compared to stale Shampoo.