Optimization over the infinite circuit sequences defining the HTMC norm

Develop a tractable optimization framework that directly optimizes over the infinite sequence of binary circuits A1, A2, … underlying the definition of the Harder than Monte Carlo (HTMC) norm ∥f∥_{H^γ}, in order to enable convex optimization of circuit size in the HTMC regime.

Background

The HTMC norm is defined via accuracy-dependent sequences of binary circuits that approximate a target function at varying precisions. The authors prove an approximate convexity of this norm for γ>2, suggesting that convex optimization could be used to minimize circuit size. However, the definition requires optimizing over an infinite sequence of circuits, a discrete structure that poses a practical barrier to convex optimization techniques.