Sharp global convergence guarantees for iterative nonconvex optimization with random data
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随机数据下迭代非凸优化的精确全局收敛保证
- 研究问题
- 对具有正态协变量的非凸回归拟合，如何从随机初始化出发，准确描述迭代算法在有限样本中的收敛速度和误差下限？
- 方法与思路
- 在每步可表示为满足一定条件的凸优化问题时，作者借助高斯比较定理，构造刻画使用样本分割算法误差的确定性序列。
- 材料陈述的贡献
- 摘要称，该序列给出精确的误差上下界；框架应用于相位恢复和混合回归，并得到多类迭代算法的全局收敛结论。
- 阅读关注点
- 核对正态设计、每步凸问题的条件、样本分割方式及总样本需求，并区分有限样本误差序列与总体极限序列。
- 阅读边界
- 来源为稿件版本未注明的 OpenAlex 索引摘要；具体条件和界未说明，不能将样本分割结论直接推广到重复使用同一数据的算法。
原文摘要
We consider a general class of regression models with normally distributed covariates, and the associated nonconvex problem of fitting these models from data. We develop a general recipe for analyzing the convergence of iterative algorithms for this task from a random initialization. In particular, provided each iteration can be written as the solution to a convex optimization problem satisfying some natural conditions, we leverage Gaussian comparison theorems to derive a deterministic sequence that provides sharp upper and lower bounds on the error of the algorithm with sample splitting. Crucially, this deterministic sequence accurately captures both the convergence rate of the algorithm and the eventual error floor in the finite-sample regime, and is distinct from the commonly used “population” sequence that results from taking the infinite-sample limit. We apply our general framework to derive several concrete consequences for parameter estimation in popular statistical models including phase retrieval and mixtures of regressions. Provided the sample size scales near linearly in the dimension, we show sharp global convergence rates for both higher-order algorithms based on alternating updates and first-order algorithms based on subgradient descent. These corollaries, in turn, reveal multiple nonstandard phenomena that are then corroborated by extensive numerical experiments.