AI 中文总结
该研究将一阶优化建模为最小时间控制问题，提出值函数作为衡量实例固有难度的基准，通过可控性指数等分析方法推导了相关特性，为一阶方法性能评估提供了新基准。
AI 中文摘要
我们将一阶优化问题建模为最小时间控制问题：迭代点作为状态，结合已观测梯度的更新作为控制，梯度范数不超过规定容差的点构成目标集。对于固定目标和起点，到达目标所需的最小神谕查询次数是一个值函数，它衡量实例的复杂度，而非某类问题的最坏情况。在强凸二次问题上，共轭梯度迭代由离散庞特里亚金条件导出，该值即为可控性指数；对于非二次问题，由海森矩阵生成的可达空间替代可控性矩阵，可行性成为可达性问题：能否到达临界点及所需步数可从该空间读取。曲率是一种资源：可验证条件表明，实例所需步数少于其极小点处二次模型的可控性指数，且该差距可随维度无界增长。因此，该值是每个实例固有难度的基准，可用于衡量任何一阶方法的性能。
英文摘要
We formulate first-order optimization as a minimum-time control problem. The iterate is the state, the update, a combination of the gradients observed so far, is the control, and the points with gradient norm at most a prescribed tolerance form the target set. For a fixed objective and start, the minimum number of oracle queries needed to reach the target is a value function: it measures the complexity of the instance, not the worst case over a class. On a strongly convex quadratic, the conjugate gradient iterates emerge from the discrete Pontryagin conditions, and the value is a controllability index. Beyond the quadratic, a Hessian-generated reachable span replaces the controllability matrix, and feasibility becomes a reachability question: whether a critical point can be reached at all, and in how many steps, is read from the span. Curvature is a resource: a checkable condition certifies that an instance needs fewer steps than the controllability index of its quadratic model at the minimizer, and the gap can grow without bound with the dimension. The value is thus a benchmark for the intrinsic difficulty of each instance, against which any first-order method can be measured.
Comments13 pages, 4 figures
