通过结构化重参数化，加速非凸优化收敛。
Sharper Convergence Rates for Nonconvex Optimisation via Reduction Mappings
- 利用已知解流形设计重参数映射，消除冗余方向。
- 改善目标函数曲率，使问题条件数更优。
- 为多种优化加速现象提供理论解释，适合算法研究者。
许多高维优化问题的极小值集具有丰富的几何结构，常因过参数化或对称性形成光滑流形。若该结构已知（至少局部），可通过减少映射重新参数化部分参数空间，使其位于解流形上。这类映射自然源于内层优化问题，有效去除冗余方向，得到低维目标函数。本文提出一个通用框架，分析此类减少如何影响优化景观。我们证明，设计良好的减少映射能改善目标函数的曲率性质，使问题条件更优，从而理论上加快梯度方法的收敛速度。该分析统一了多种利用最优性处结构信息加速收敛的情形，为实际优化算法中观察到的性能提升提供了原则性解释。
原文摘要 · Abstract (English)
Many high-dimensional optimisation problems exhibit rich geometric structures in their set of minimisers, often forming smooth manifolds due to over-parametrisation or symmetries. When this structure is known, at least locally, it can be exploited through reduction mappings that reparametrise part of the parameter space to lie on the solution manifold. These reductions naturally arise from inner optimisation problems and effectively remove redundant directions, yielding a lower-dimensional objective. In this work, we introduce a general framework to understand how such reductions influence the optimisation landscape. We show that well-designed reduction mappings improve curvature properties of the objective, leading to better-conditioned problems and theoretically faster convergence for gradient-based methods. Our analysis unifies a range of scenarios where structural information at optimality is leveraged to accelerate convergence, offering a principled explanation for the empirical gains observed in such optimisation algorithms.
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