Is Softmax Loss all you need? A Principled Analysis of Softmax-family Loss

The **Softmax Cross-Entropy** Loss is one of the most widely employed surrogate objectives for classification and ranking tasks. To elucidate its theoretical properties, the **Fenchel-Young** framework situates it as a canonical instance within a broad family of surrogates. Concurrently, another line of research has addressed scalability when the number of classes is exceedingly large, proposing alternatives that reduce the computational cost of the exact objective. Building on these two perspectives, we present a principled investigation of **Softmax-family** losses. We examine their consistency with task-specific classification and ranking metrics, characterize their gradient structures and score-space smoothness, and derive convergence bounds in a regularized fixed-feature setting. We also analyze approximate methods through a loss-level bias–variance decomposition and per-epoch computational costs, clarifying trade-offs between approximation fidelity and efficiency. Experiments across representative recommendation backbones reveal setting-dependent differences in predictive performance and training behavior, complementing the theoretical analysis. Together, these results establish a principled basis for comparing surrogate objectives and offer practical guidance for loss selection in large-scale machine learning.