Reproduce “Learning Theory for Kernel Bilevel Optimization” on OpenResearch

About this paper

Fares El Khoury, Edouard Pauwels, Samuel Vaiter, Michael Arbel

The authors analyze generalization in kernel bilevel optimization, where an inner RKHS model is evaluated using separate outer data. They show that the empirical value function and gradient approximate their population counterparts at expected rate O(m^-1/2+n^-1/2) on compact parameter sets, under fixed regularization and stated assumptions. A gradient-descent guarantee adds an iteration term, but does not guarantee a global optimum.

Key insights

• The gradient of the empirical KBO value function is shown to be identical to a direct plug-in estimator, which simplifies its statistical analysis by operating on fixed RKHS-valued objects.
• Generalization error for KBO's value function and gradient decays at a parametric rate of $O(1/\sqrt{m} + 1/\sqrt{n})$, indicating that both inner-level ($n$) and outer-level ($m$) sample sizes contribute equally to controlling statistical error.
• The analysis of gradient-based algorithms applied to empirical KBO reveals a direct trade-off between the statistical error, determined by sample sizes, and the optimization error, dependent on the number of iterations.

Summary by alphaXiv.