Behavior at the critical “pentuple point” α = β = 1/2

Characterize the behavior of the loss curve and compute-optimal frontier at the critical point α = β = 1/2 in the (α, β)-phase diagram for the PLRF model trained with one-pass SGD, where all forcing and kernel components (Fpp, Fac, F0, Kpp) mix and interact.

Background

The authors map out four phases (+ subphases) of compute-optimal behavior depending on data complexity α and target complexity β. Away from phase boundaries, distinct terms in the forcing and kernel functions dominate the loss and determine scaling exponents.

At the critical point α = β = 1/2—dubbed the pentuple point—all components contribute and interact, making the dynamics nontrivial. The authors highlight this as an unresolved case where a complete characterization remains to be established.