Existence of subgame-perfect versions of ITEM and OGM-G

Establish the existence of subgame-perfect first-order methods corresponding to the Information-Theoretic Exact Method (ITEM) for smooth strongly convex optimization and to OGM-G for minimizing the gradient norm; specifically, construct algorithms in these two settings whose strategies satisfy the subgame perfect equilibrium criterion by dynamically adapting to observed first-order information.

Background

The paper introduces the Subgame Perfect Gradient Method (SPGM) and a game-theoretic framework that strengthens classical minimax optimality by requiring subgame perfect equilibrium behavior. SPGM is shown to achieve dynamic, history-dependent guarantees that are optimal among gradient-span methods for unconstrained L-smooth convex minimization.

In the conclusion, the authors propose extending the subgame-perfect approach beyond the unconstrained smooth convex objective-gap setting to other well-studied contexts. ITEM is a minimax-optimal method for smooth strongly convex minimization, and OGM-G targets minimizing the gradient norm for smooth convex problems. The conjecture asks for analogous subgame-perfect algorithms in these two settings.