PDE-Based Bounds on Optimal Exponents in Active Simple Hypothesis Testing

We present general upper and lower bounds on the optimal error exponent for the Active Simple Hypothesis Testing (ASHT) problem. ASHT has been extensively studied across diverse fields including Information Theory, Bandits, and Operations Research, a precise characterization of this optimal exponent has remained elusive. This work combines a recent Partial Differential Equation (PDE) characterization of the optimal exponent by Vijayan (2025) with the comparison principle of viscosity solutions of Hamilton-Jacobi PDEs. This leads to bounds that are parametrized by convex functions, sets and lower Hamiltonian relaxations, a flexibility that generalizes many known results in the literature.