Gradient-based algorithms are central to modern statistical estimation, yet their statistical analysis is often restricted to fixed-time behavior, such as convergence to a population target or fluctuations at a prescribed iteration. In many applications, however, uncertainty quantification is needed along the entire optimization path, especially when the stopping time is data-dependent or divergent. In this paper, we develop a theory for time-uniform statistical inference on gradient flows arising from empirical risk minimization. We prove a uniform central limit theorem that characterizes the deviation between empirical and population gradient flows as a continuous-time Gaussian process over the entire nonnegative real line. Building on this result, we introduce an algorithm-aware covariance estimator that evolves jointly with the gradient flow and avoids matrix inversion, resampling, or sample splitting. We show that the covariance estimator is uniformly consistent over time and use it to construct confidence intervals for the target parameter with asymptotically valid coverage. Our results connect optimization dynamics with statistical inference and provide practical tools for uncertainty quantification in gradient-based methods.

翻译：基于梯度的算法是现代统计估计的核心，然而对其的统计分析通常局限于固定时刻的行为，例如收敛到总体目标或指定迭代次数的波动。然而在许多应用中，整个优化路径上的不确定性量化是必需的，特别是当停止时间依赖于数据或发散时。本文提出了一种基于经验风险最小化产生的梯度流的时间一致统计推断理论。我们证明了一个一致中心极限定理，该定理将经验梯度流与总体梯度流之间的偏差刻画为整个非负实数轴上的连续时间高斯过程。基于此结果，我们引入了一种与梯度流协同演化的算法感知协方差估计器，避免了矩阵求逆、重采样或样本分割。我们证明了该协方差估计器随时间一致收敛，并利用它构建目标参数的置信区间，其覆盖概率渐近有效。我们的结果将优化动力学与统计推断联系起来，为基于梯度方法的不确定性量化提供了实用工具。