Vein recognition, valued for its contactless, anti-counterfeiting, and interference-resistant properties, has become a key focus in identity authentication. In deep learning-based vein recognition, the loss function plays a vital role in guiding model optimization and achieving high recognition accuracy. However, existing methods, such as Poly-1, rely on fixed, manually tuned hyperparameters, which limit their adaptability to different data distributions and network architectures. To address this issue, this paper proposes LACE (Learnable Adaptive Cross-Entropy Loss), a novel adaptive loss function based on the Taylor expansion of the standard cross-entropy, which incorporates learnable parameters to dynamically optimize the polynomial coefficients. Unlike Poly-1, LACE automatically adjusts its parameters during training, better adapting to diverse scenarios. Experiments on three vein datasets and nine deep learning models demonstrate that LACE consistently improves recognition performance, reducing the average equal error rates by 0.15%, 0.16%, and 0.37% on FV_USM, TJU_PV600, and VERA_PV200, respectively.