A 75/61 Upper Bound on the Universal Price of Three-Arm Gaussian Best-Arm Identification

Abstract

For fixed-budget best-arm identification with three unit-variance Gaussian arms, this preprint gives a constructive upper bound on the universal price of best-arm identification: P₃ ≤ 75/61 = 1.229508196721312… The proposed two-stage soft-freeze policy uses a uniform pilot, freezes additional sampling of the pilot-worst arm while retaining it for final recommendation, and reallocates the remaining budget to the other two arms. The proof establishes a uniform large-deviation exponent bound via six exact convex Gaussian quadratic projections and an exact Bernstein-basis positivity certificate for a quintic polynomial. The result narrows the previously known explicit interval for P₃ by about 55.4% in width. This work does not determine P₃ exactly.