Semiparametric Efficient Bilevel Gradient Estimation

Functional bilevel methods estimate a lower-level function and plug it into a hypergradient, but this plug-in gradient can retain first-order bias when the lower-level problem is learned nonparametrically. To remove this bias, we develop a semiparametric debiasing theory for population bilevel gradients based on the efficient influence function. This perspective leads to a cross-fitted orthogonal hypergradient estimator for which we establish asymptotic normality together with uniform control over the outer parameter. Under quadratic losses, the estimator reduces to a simple doubly robust score based on conditional mean nuisances. On synthetic bilevel benchmarks with known ground truth, the method tracks the oracle efficient-gradient benchmark and improves over plug-in functional hypergradients and regularized kernel bilevel baselines.

Learning Theory for Kernel Bilevel Optimization

Bilevel optimization has emerged as a technique for addressing a wide range of machine learning problems that involve an outer objective implicitly determined by the minimizer of an inner problem. In this paper, we investigate the generalization properties for kernel bilevel optimization problems where the inner objective is optimized over a Reproducing Kernel Hilbert Space. This setting enables rich function approximation while providing a foundation for rigorous theoretical analysis. In this context, we establish novel generalization error bounds for the bilevel problem under finite-sample approximation. Our approach adopts a functional perspective, inspired by (Petrulionyte et al., 2024), and leverages tools from empirical process theory and maximal inequalities for degenerate $U$-processes to derive uniform error bounds. These generalization error estimates allow to characterize the statistical accuracy of gradient-based methods applied to the empirical discretization of the bilevel problem.