Riemannian Bilevel Optimization with Gradient Aggregation

Zhuo Chen; Xinjian Xu; Shihui Ying; Tieyong Zeng

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arXiv · 2510.15305

Riemannian Bilevel Optimization with Gradient Aggregation

Zhuo Chen·Xinjian Xu·Shihui Ying·Tieyong Zeng

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Abstract

We study bilevel optimization on Riemannian manifolds when the lower-level solution set is a positive-dimensional submanifold, so that implicit differentiation fails. We propose Riemannian Bilevel Descent Aggregation (RBDA), which extends bilevel descent aggregation to manifolds. Its inner loop aggregates the lower-level descent direction with the upper-level gradient under a decaying multiplier, and its hypergradient is the reverse-mode derivative of the unrolled loop. Under geodesic convexity and quadratic growth of the lower level, the inner iterates converge to a point of the optimistic solution set at a polynomial rate. Approximate minimizers of the objective with a finite number of inner iterations converge to minimizers of the optimistic value. In the experiments RBDA selects the optimistic solution where the implicit and unrolled estimators remain at the initial point or stop at a larger query loss. While each of its outer steps costs more than that of the unrolled estimator, it attains the highest test accuracy in data hyper-cleaning.

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BibTeX RIS

Zhuo Chen, Xinjian Xu, Shihui Ying, Tieyong Zeng. 2026-09-14. Riemannian Bilevel Optimization with Gradient Aggregation. https://arxiv.org/abs/2510.15305

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