Fully First-Order Algorithms for Online Bilevel Optimization
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910204182921216 |
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| author | Jia, Tingkai Chen, Cheng |
| author_facet | Jia, Tingkai Chen, Cheng |
| contents | In this work, we study nonconvex-strongly convex online bilevel optimization (OBO) using only first-order oracle. Existing OBO algorithms are mainly based on hypergradient descent, which requires access to a Hessian-vector product (HVP) oracle and potentially incurs high computational costs. By reformulating the original OBO problem as a single-level online problem with inequality constraints and constructing a sequence of Lagrangian function, we eliminate the need for HVPs arising from implicit differentiation. Specifically, we propose a fully first-order algorithm for OBO, and provide theoretical guarantees showing that it achieves regret of $O(1 + V_T + H_{2,T})$ with a total of $O(T\log T)$ iterations, where $V_T$ measures the variation in function values and $H_{2,T}$ characterizes the drift variation of the inner-level optimal solution. We also establish a sublinear regret bound under the single-loop structure by introducing additional gradient-variation terms. Furthermore, we develop an improved variant with an adaptive inner-iteration scheme, which removes the dependence on $H_{2,T}$ and achieves regret of $O(\log T + V_T)$. Finally, under the stochastic OBO setting, we establish the regret bound for the fully first-order algorithm, i.e., $O(T^{2/3}(1 + σ^2) + V_T + H_{2,T})$. Numerical experiments demonstrate the feasibility of our algorithm and support our theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11665 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fully First-Order Algorithms for Online Bilevel Optimization Jia, Tingkai Chen, Cheng Machine Learning Optimization and Control In this work, we study nonconvex-strongly convex online bilevel optimization (OBO) using only first-order oracle. Existing OBO algorithms are mainly based on hypergradient descent, which requires access to a Hessian-vector product (HVP) oracle and potentially incurs high computational costs. By reformulating the original OBO problem as a single-level online problem with inequality constraints and constructing a sequence of Lagrangian function, we eliminate the need for HVPs arising from implicit differentiation. Specifically, we propose a fully first-order algorithm for OBO, and provide theoretical guarantees showing that it achieves regret of $O(1 + V_T + H_{2,T})$ with a total of $O(T\log T)$ iterations, where $V_T$ measures the variation in function values and $H_{2,T}$ characterizes the drift variation of the inner-level optimal solution. We also establish a sublinear regret bound under the single-loop structure by introducing additional gradient-variation terms. Furthermore, we develop an improved variant with an adaptive inner-iteration scheme, which removes the dependence on $H_{2,T}$ and achieves regret of $O(\log T + V_T)$. Finally, under the stochastic OBO setting, we establish the regret bound for the fully first-order algorithm, i.e., $O(T^{2/3}(1 + σ^2) + V_T + H_{2,T})$. Numerical experiments demonstrate the feasibility of our algorithm and support our theoretical findings. |
| title | Fully First-Order Algorithms for Online Bilevel Optimization |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2602.11665 |