Biometric recognition has primarily addressed closed-set identification, assuming all probe subjects are known to be in the gallery. However, most practical applications involve open-set biometrics, where probe subjects may or may not be present in the gallery. This poses distinct challenges in effectively distinguishing individuals in the gallery while minimizing false detections. Despite assuming that powerful biometric models can excel in both closed- and open-set scenarios, existing loss functions are inconsistent with open-set evaluation. The genuine (mated) and imposter (non-mated) similarity scores symmetrically and neglect the relative magnitudes of imposter scores. To address these issues, we introduce novel loss functions: (1) the \textbf{identification-detection} loss optimized for open-set performance under selective thresholds and (2) \textbf{relative threshold minimization} to reduce the maximum negative score for each probe. Across diverse biometric tasks, including face recognition, gait recognition, and person re-identification, our experiments demonstrate the effectiveness of the proposed loss functions, significantly enhancing open-set performance while positively impacting closed-set performance. Upon publication, we will release our code and models.