Entropic soft-min operators are widely used to obtain smooth approximations of minimum and argmin mechanisms in optimization, machine learning, and reinforcement learning. The quality of this approximation is controlled by an inverse temperature parameter that governs the trade-off between smoothness and fidelity, yet its selection is usually based on global heuristics or worst-case bounds that do not account for the geometry of the candidate cost vector. This study investigates the calibration of the inverse temperature parameter from a geometry-aware perspective, with explicit guarantees on the approximation error between the entropic soft-min and the exact minimum value. After establishing the structural properties of the relaxation error, including monotonicity with respect to the inverse temperature and its dependence on the geometry of the near-optimal set, we introduce a conformal calibration rule that selects the smallest inverse temperature, ensuring that a prescribed upper quantile of the approximation error remains below a target tolerance with distribution-free finite-sample validity. The resulting selector adapts to the geometry distribution represented in the calibration population and provides a principled alternative to mean-based and worst-case tuning rules. Numerical experiments, including geometry-controlled benchmarks and a contextual bandit setting illustrating the impact of geometry-aware calibration on decision-making under estimated action values, show that the proposed method accurately tracks oracle calibration temperatures, preserves the desired operator-level coverage, and makes explicit how geometric heterogeneity governs the effective sharpness required by the soft-min approximation. Additional shifted evaluations illustrate the role of exchangeability in the validity guarantee and the consequences of transferring temperatures across populations with different near-optimal geometries.
Solanes et al. (Fri,) studied this question.