$\alpha$-Divergence Loss Function for Neural Density Ratio Estimation
<|reference_start|>$\alpha$-Divergence Loss Function for Neural Density Ratio Estimation: Density ratio estimation (DRE) is a fundamental machine learning technique for capturing relationships between two probability distributions. State-of-the-art DRE methods estimate the density ratio using neural networks trained with loss functions derived from variational representations of $f$-divergence. However, existing methods face optimization challenges, such as overfitting due to lower-unbounded loss functions, biased mini-batch gradients, vanishing training loss gradients, and high sample requirements for Kullback-Leibler (KL) divergence loss functions. To address these issues, we focus on $\alpha$-divergence, which provides a suitable variational representation of $f$-divergence. Subsequently, a novel loss function for DRE, the $\alpha$-divergence loss function ($\alpha$-Div), is derived. $\alpha$-Div is concise but offers stable and effective optimization for DRE. The boundedness of $\alpha$-divergence provides the potential for successful DRE with data exhibiting high KL-divergence. Our numerical experiments demonstrate the effectiveness in optimization using $\alpha$-Div. However, the experiments also show that the proposed loss function offers no significant advantage over the KL-divergence loss function in terms of RMSE for DRE. This indicates that the accuracy of DRE is primarily determined by the amount of KL-divergence in the data and is less dependent on $\alpha$-divergence.<|reference_end|>
@article{kitazawa2024$\alpha$-divergence,
    title={$\alpha$-Divergence Loss Function for Neural Density Ratio Estimation},
    author={Yoshiaki Kitazawa},
    journal={arXiv preprint arXiv:2402.02041},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.02041},
    primaryClass={stat.ML cs.LG}
}