# Statistical Inference on Gradient Flows

arXiv: https://arxiv.org/abs/2606.01257 · HTML: https://mlsanity.com/paper/2606.01257/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 78 references that are in the map.
- First posted on arXiv 31 May 2026.

## Builds on (most cited first, 30 of 46)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [The Landscape of Empirical Risk for Non-convex Losses](https://mlsanity.com/paper/1607.06534/) (2016)
- [Robust Estimation via Robust Gradient Estimation](https://mlsanity.com/paper/1802.06485/) (2018)
- [Statistical Inference for Model Parameters in Stochastic Gradient Descent](https://mlsanity.com/paper/1610.08637/) (2016)
- [High-dimensional limit theorems for SGD: Effective dynamics and critical scaling](https://mlsanity.com/paper/2206.04030/) (2022)
- [Singularity, Misspecification, and the Convergence Rate of EM](https://mlsanity.com/paper/1810.00828/) (2018)
- [Online Covariance Matrix Estimation in Stochastic Gradient Descent](https://mlsanity.com/paper/2002.03979/) (2020)
- [Symmetries, flat minima, and the conserved quantities of gradient flow](https://mlsanity.com/paper/2210.17216/) (2022)
- [Statistical consistency and asymptotic normality for high-dimensional robust M-estimators](https://mlsanity.com/paper/1501.00312/) (2015)
- [Differentially private inference via noisy optimization](https://mlsanity.com/paper/2103.11003/) (2021)
- [Online Statistical Inference for Stochastic Optimization via Kiefer-Wolfowitz Methods](https://mlsanity.com/paper/2102.03389/) (2021)
- [Gaussian Approximation and Multiplier Bootstrap for Stochastic Gradient Descent](https://mlsanity.com/paper/2502.06719/) (2025)
- [Gradient flows and proximal splitting methods: A unified view on accelerated and stochastic optimization](https://mlsanity.com/paper/1908.00865/) (2019)
- [Risk Comparisons in Linear Regression: Implicit Regularization Dominates Explicit Regularization](https://mlsanity.com/paper/2509.17251/) (2025)
- [Statistical Inference with Stochastic Gradient Methods under φ-mixing Data](https://mlsanity.com/paper/2302.12717/) (2023)
- [High-Dimensional Analysis of Gradient Flow for Extensive-Width Quadratic Neural Networks](https://mlsanity.com/paper/2601.10483/) (2026)
- [Path Length Bounds for Gradient Descent and Flow](https://mlsanity.com/paper/1908.01089/) (2019)
- [SGD with Dependent Data: Optimal Estimation, Regret, and Inference](https://mlsanity.com/paper/2601.01371/) (2026)
- [Precise gradient descent training dynamics for finite-width multi-layer neural networks](https://mlsanity.com/paper/2505.04898/) (2025)
- [Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces](https://mlsanity.com/paper/2304.05239/) (2023)
- [Limit Theorems for Stochastic Gradient Descent with Infinite Variance](https://mlsanity.com/paper/2410.16340/) (2024)
- [Long-time dynamics and universality of nonconvex gradient descent](https://mlsanity.com/paper/2509.11426/) (2025)
- [Improved Central Limit Theorem and Bootstrap Approximations for Linear Stochastic Approximation](https://mlsanity.com/paper/2510.12375/) (2025)
- [A Researcher's Guide to Empirical Risk Minimization](https://mlsanity.com/paper/2602.21501/) (2026)
- [Optimal One-pass Nonparametric Estimation Under Memory Constraint](https://mlsanity.com/paper/2208.08668/) (2022)
- [Uncertainty quantification for iterative algorithms in linear models with application to early stopping](https://mlsanity.com/paper/2404.17856/) (2024)
- [Asymptotic Time-Uniform Inference for Parameters in Averaged Stochastic Approximation](https://mlsanity.com/paper/2410.15057/) (2024)

## Built on by (most cited first, 1 of 1)

- [Estimating Population-Risk Curves Along Nonconvex Gradient Flows from the Training Sample](https://mlsanity.com/paper/2608.30261/) (2026)

## Abstract (arXiv)

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.