## 1 Introduction
In machine learning, the bias-variance dilemma emerges from the endeavor to minimize bias and variance simultaneously. Complex models often possess low bias and high variance, whereas relatively simple models tend to have high bias and low variance. This predicament encapsulates the challenge of enabling a model to grasp the optimal input-output relationship while maintaining the capacity to generalize beyond the original training data samples. Model ensembles, much like voting systems, involve each member contributing equally through predictive votes [12]. This incorporation of varied perspectives fosters diversity within the ensemble, ultimately leading to diminished variance and enhanced generalization capabilities beyond training data. The voting regressor utilizes the ensembling concept by averaging individual predictions for a conclusive outcome. Facial-beauty data may exhibit biases, which can affect data distribution. For example, there may be a bias towards images of certain beauty scores, ethnicities, or age groups which can affect the generalizability of the model trained on the dataset. The integration of multiple loss functions in a linear combination has proven to enhance results, as supported by previous studies [13; 14; 15]. Thus, the most effective solution in addressing these biases and ensuring the generalizability of the model is using a combination of loss functions to integrate diverse data-driven techniques and leverage their benefits. This work represents a pioneering effort at incorporating the Log-cosh loss function into FBP, combined with L1 and L2 losses, using ensemble average in a CNN-regression-based model for predicting facial-image beauty scores. This distinctive contribution highlights the originality and uniqueness of our work in the field of FBP through ensembling the average of L2, L1, and Log-cosh regression-loss functions within a deep network. Accordiningly, the model can better capture the underlying patterns and relationships in the data, leading to improved generalization and robustness in FBP across various demographic factors, such as gender, age, and ethnicity. Recognizing the significance of improving prediction reliability and precision, the fusion of diverse data-driven techniques into an ensemble has become a prominent research domain in recent times. The key contributions of this paper are as follows:  
1. Leveraging the Log-cosh loss function within the context of FBP to enhance the learning process; to the best of our knowledge, we are the first to use it in quantifying the beauty in facial images. 2. Refining the performance of three distinct pretrained CNNs, namely AlexNet, VGG16, and FIAC-Net, for the purpose of estimating the beauty score within facial images. This enhancement is achieved through the process of tuning and retraining these networks on separate regression-loss functions, namely L1 loss, L2 loss, and Log-cosh, on an individual basis for each network. 3. Proposing a new ensemble average cost function that effectively combines L2, L1, and Log-cosh loss functions to enhance the model's generalization and robustness in predicting the beauty scores across subjects with diverse gender, age, and ethnic characteristics, and further integrating this ensemble cost function with diverse CNN models demonstrate its efficacy in enhancing the capabilities of various deep-learning architectures for the FBP task. 4. Utilizing distinct FBP benchmarks with the aim of comparing the performance of the proposed models in both restricted environments, represented by the SCUT-FBP and SCUT-FBP5500 datasets, and wild facial images captured under unconstrained conditions, as represented by the MEBeauty dataset. This paper is structured in a manner that includes a brief overview of the relevant studies in Section 2, followed by a description of the suggested framework in Section 3. The empirical findings are presented in Section 4, while Section 5 provides a concluding summary of the study.