🤖 AI Summary
Existing neural density ratio estimation (DRE) methods employing KL-divergence loss suffer from overfitting and training instability due to the loss’s unboundedness below, vanishing gradients, mini-batch bias, and sensitivity to sample size. This work introduces, for the first time, the α-divergence family into DRE loss design. Leveraging its f-divergence variational representation, we construct the α-Div loss—a bounded, differentiable objective that simultaneously ensures loss boundedness and gradient stability. The density ratio is parameterized by a neural network and optimized via stochastic gradient descent. Experiments demonstrate that α-Div significantly improves training stability and convergence speed, maintaining effective optimization even under high KL divergence between distributions. Its root-mean-square error (RMSE) accuracy matches that of KL-loss-based methods, indicating that the fundamental accuracy limit stems from intrinsic data properties rather than the choice of loss function.
📝 Abstract
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 $f$-divergences. 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 of $alpha$-Div in optimization. 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.