Cost-Aware AUC Optimization via Adaptive Kernel Density Estimation

The Area Under the Receiver Operating Characteristics Curve (AUC) is a widely used metric for evaluating model performance across all possible decision thresholds. Existing methods for AUC optimization typically assume a predefined parametric distribution of thresholds. However, the optimal decision threshold depends on the misclassification costs, which follow a non-parametric distribution.This motivates us to introduce a variant of AUC, termed Cost-aware AUC (CAUC), where the thresholds are conditioned on an empirically determined cost distribution. Unfortunately, as a bilevel problem, it is challenging to directly optimize the CAUC: 1) The inner problem of finding the optimal thresholds is non-convex, leading to potential issues with convergence; 2) The outer problem involves the derivative of False Positive Rate (FPR) w.r.t. the threshold, which is unavailable without an explicit formulation of threshold distribution. To address challenge 1), we utilize the convex relaxation technique to reshape the inner problem into a convex one. Facing challenge 2), we propose an adaptive kernel density estimation framework. Specifically, the derivative of FPR is considered an aggregation of various kernel functions. To avoid manually crafting the aggregation function, we propose a finite-difference-based stochastic algorithm to optimize the model without explicit aggregation function. Theoretically, the proposed algorithm enjoys a convergence rate of $\mathcal {O}(\epsilon ^{-4})$O(ε-4). Empirical studies across various datasets and cost distributions speak to the effectiveness and soundness of our framework.