Towards Nonlinear Sparse AUC Maximization via Compositional Stochastic Hard Thresholding

Abstract

The Area Under the ROC Curve (AUC) is an important evaluation metric for both linear and, in particular, nonlinear classification models, owing to its robustness against class imbalance. Sparse learning with an ℓ₀ constraint can enhance model interpretability and generalization. Prior work has shown that, in the linear setting, the pairwise formulation of AUC maximization can be reformulated as a standard pointwise empirical risk minimization problem, which enables efficient optimization using hard-thresholding gradient descent for ℓ₀-constrained AUC maximization. Extending this approach to the nonlinear setting remains largely unexplored, even though we establish that pairwise AUC maximization in this setting is equivalent to a pointwise compositional optimization problem; however, designing a compositional optimization algorithm compatible with hard-thresholding operators remains an open challenge. To address this challenge, in this paper, we propose a novel algorithm—Compositional Stochastic Hard Thresholding (CSHT)—for nonlinear sparse AUC maximization. Specifically, CSHT integrates stochastic variance-reduced gradient techniques with hard-thresholding projections to effectively reduce gradient estimation variance while enforcing sparsity. Notably, we provide a rigorous convergence analysis and prove that CSHT achieves linear convergence up to a tolerance bound. To the best of our knowledge, this is the first stochastic hard-thresholding algorithm tailored for nonlinear sparse AUC maximization. Extensive experiments on (a) nonlinear sparse AUC maximization using Random Fourier Feature-based kernel approximation and (b) universal adversarial attack scenarios demonstrate the superior performance of CSHT over existing methods, attributed to its unified treatment of nonlinearity and sparsity.