Binno: A 1st-order method for Bi-level Nonconvex Nonsmooth Optimization for Matrix Factorizations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Selicato, Laura Esposito, Flavia Ang, Andersen |
| author_facet | Selicato, Laura Esposito, Flavia Ang, Andersen |
| contents | Nonconvex and nonsmooth bi-level optimization poses critical theoretical challenges, while arising in several applications. In this work, we develop a method for nonconvex, nonsmooth bi-level optimization and introduce Binno, a first-order method that builds on proximal-gradient updates within the the proximal alternate minimization framework with descent conditions from variational analysis. Binno couples the two levels via a descent-driven averaging mechanism, extending single-level proximal schemes to the nonconvex nonsmooth bi-level setting. We show thatBinno induces a descent property for a suitable surrogate of the bi-level objective. Each iteration performs blockwise proximal-gradient updates for the upper and lower problems separately, then forms a calibrated, block-diagonal convex combination of the two iterates. A linesearch selects combination weights enforcing simultaneous descent of both objectives. We give conditions ensuring that both these weights and the descent directions induced by the associated proximal-gradient maps exist. We apply Binno to sparse low-rank factorization, where the upper level uses elementwise l1 penalties and the lower level uses nuclear norms, coupled via a Frobenius data term. We test Binno on synthetic matrices and a real traffic-video dataset, achieving lower reconstruction error and higher peak signal-to-noise ratio than standard methods. We also validate it on a regularized market-clearing problem, where it selects policy-preferred equilibria, and compare it with bi-level baseline, showing consistent improvements |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21390 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Binno: A 1st-order method for Bi-level Nonconvex Nonsmooth Optimization for Matrix Factorizations Selicato, Laura Esposito, Flavia Ang, Andersen Optimization and Control 65K10, 90C26, 90C30 Nonconvex and nonsmooth bi-level optimization poses critical theoretical challenges, while arising in several applications. In this work, we develop a method for nonconvex, nonsmooth bi-level optimization and introduce Binno, a first-order method that builds on proximal-gradient updates within the the proximal alternate minimization framework with descent conditions from variational analysis. Binno couples the two levels via a descent-driven averaging mechanism, extending single-level proximal schemes to the nonconvex nonsmooth bi-level setting. We show thatBinno induces a descent property for a suitable surrogate of the bi-level objective. Each iteration performs blockwise proximal-gradient updates for the upper and lower problems separately, then forms a calibrated, block-diagonal convex combination of the two iterates. A linesearch selects combination weights enforcing simultaneous descent of both objectives. We give conditions ensuring that both these weights and the descent directions induced by the associated proximal-gradient maps exist. We apply Binno to sparse low-rank factorization, where the upper level uses elementwise l1 penalties and the lower level uses nuclear norms, coupled via a Frobenius data term. We test Binno on synthetic matrices and a real traffic-video dataset, achieving lower reconstruction error and higher peak signal-to-noise ratio than standard methods. We also validate it on a regularized market-clearing problem, where it selects policy-preferred equilibria, and compare it with bi-level baseline, showing consistent improvements |
| title | Binno: A 1st-order method for Bi-level Nonconvex Nonsmooth Optimization for Matrix Factorizations |
| topic | Optimization and Control 65K10, 90C26, 90C30 |
| url | https://arxiv.org/abs/2510.21390 |