Broximal Alignment for Global Non-Convex Optimization
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915937997815808 |
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| author | Gruntkowska, Kaja Li, Hanmin Qian, Xun Richtárik, Peter |
| author_facet | Gruntkowska, Kaja Li, Hanmin Qian, Xun Richtárik, Peter |
| contents | Most non-convex optimization theory is built around gradient dynamics, leaving global convergence largely unexplored. The dominant paradigm focuses on stationarity, certifying only that the gradient norm vanishes, which is often a weak proxy for actual optimization success. In practice, gradient norms can stagnate or even increase during training, and stationary points may be far from global solutions. In this work, we propose a new framework for global non-convex optimization that avoids gradient-based reasoning altogether. We revisit the Ball Proximal Point Method (BPM), a trust-region-style algorithm introduced by Gruntkowska et al. (2025), and propose a novel structural condition - Broximal Alignment - under which BPM provably converges to a global minimizer. Our condition requires no convexity, smoothness, or Lipschitz assumptions, and it permits multiple and disconnected global minima as well as non-optimal local minima. We show that this class generalizes standard non-convex frameworks such as quasiconvexity, star convexity, quasar convexity, and the aiming condition. Our results provide a new conceptual foundation for global non-convex optimization beyond stationarity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13483 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Broximal Alignment for Global Non-Convex Optimization Gruntkowska, Kaja Li, Hanmin Qian, Xun Richtárik, Peter Optimization and Control Most non-convex optimization theory is built around gradient dynamics, leaving global convergence largely unexplored. The dominant paradigm focuses on stationarity, certifying only that the gradient norm vanishes, which is often a weak proxy for actual optimization success. In practice, gradient norms can stagnate or even increase during training, and stationary points may be far from global solutions. In this work, we propose a new framework for global non-convex optimization that avoids gradient-based reasoning altogether. We revisit the Ball Proximal Point Method (BPM), a trust-region-style algorithm introduced by Gruntkowska et al. (2025), and propose a novel structural condition - Broximal Alignment - under which BPM provably converges to a global minimizer. Our condition requires no convexity, smoothness, or Lipschitz assumptions, and it permits multiple and disconnected global minima as well as non-optimal local minima. We show that this class generalizes standard non-convex frameworks such as quasiconvexity, star convexity, quasar convexity, and the aiming condition. Our results provide a new conceptual foundation for global non-convex optimization beyond stationarity. |
| title | Broximal Alignment for Global Non-Convex Optimization |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2604.13483 |