Explicit formula for the equilibrium time t*

Derive an explicit general formula for the equilibrium time t*(w, σ^2/ε, h1, T1, ..., hn, Tn), the implicitly defined mapping used to set computation/communication budgets in Shadowheart SGD and to express both the algorithm’s upper time-complexity bound and the corresponding lower bound for centralized methods with compressed communication under arbitrary device heterogeneity.

Background

Shadowheart SGD’s time complexity bound is expressed in terms of an implicitly defined equilibrium time t*(w, σ^2/ε, h1, T1, ..., hn, Tn). The mapping t* arises from balancing computation and communication across heterogeneous workers and is currently specified via a minimax construction coupled with solving an equation for s*(j).

The authors establish existence (well-definedness) of t* and show it also appears in the lower bound for all centralized methods with compression, indicating its fundamental nature. However, they acknowledge that only an implicit definition is provided, leaving open whether a more explicit expression can be obtained in general.