# Asymptotic Time-Uniform Inference for Parameters in Averaged Stochastic Approximation

arXiv: https://arxiv.org/abs/2410.15057 · HTML: https://mlsanity.com/paper/2410.15057/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 56 references that are in the map.
- First posted on arXiv 19 Oct 2024.

## Builds on (most cited first, 22 of 22)

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Statistical Inference for Model Parameters in Stochastic Gradient Descent](https://mlsanity.com/paper/1610.08637/) (2016)
- [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021)
- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [Online Covariance Matrix Estimation in Stochastic Gradient Descent](https://mlsanity.com/paper/2002.03979/) (2020)
- [Testing exchangeability: fork-convexity, supermartingales, and e-processes](https://mlsanity.com/paper/2102.00630/) (2021)
- [Time-uniform central limit theory and asymptotic confidence sequences](https://mlsanity.com/paper/2103.06476/) (2021)
- [A Statistical Analysis of Polyak-Ruppert Averaged Q-learning](https://mlsanity.com/paper/2112.14582/) (2021)
- [Fast and Robust Online Inference with Stochastic Gradient Descent via Random Scaling](https://mlsanity.com/paper/2106.03156/) (2021)
- [Sequential estimation of quantiles with applications to A/B-testing and best-arm identification](https://mlsanity.com/paper/1906.09712/) (2019)
- [Online Statistical Inference for Nonlinear Stochastic Approximation with Markovian Data](https://mlsanity.com/paper/2302.07690/) (2023)
- [Statistical Inference for Online Decision-Making: In a Contextual Bandit Setting](https://mlsanity.com/paper/2010.07283/) (2020)
- [Statistical Inference for Online Decision Making via Stochastic Gradient Descent](https://mlsanity.com/paper/2010.07341/) (2020)
- [Online Statistical Inference for Stochastic Optimization via Kiefer-Wolfowitz Methods](https://mlsanity.com/paper/2102.03389/) (2021)
- [HiGrad: Uncertainty Quantification for Online Learning and Stochastic Approximation](https://mlsanity.com/paper/1802.04876/) (2018)
- [On Constructing Confidence Region for Model Parameters in Stochastic Gradient Descent via Batch Means](https://mlsanity.com/paper/1911.01483/) (2019)
- [Time-Uniform Self-Normalized Concentration for Vector-Valued Processes](https://mlsanity.com/paper/2310.09100/) (2023)
- [Online Statistical Inference for Contextual Bandits via Stochastic Gradient Descent](https://mlsanity.com/paper/2212.14883/) (2022)
- [SGMM: Stochastic Approximation to Generalized Method of Moments](https://mlsanity.com/paper/2308.13564/) (2023)

## Built on by (most cited first, 3 of 3)

- [Federated Learning of Quantile Inference under Local Differential Privacy](https://mlsanity.com/paper/2509.21800/) (2025)
- [Statistical Inference on Gradient Flows](https://mlsanity.com/paper/2606.01257/) (2026)
- [Sharp asymptotic theory for Q-learning with LDTZ learning rate and its generalization](https://mlsanity.com/paper/2604.04218/) (2026)

## Abstract (arXiv)

We study time-uniform statistical inference for parameters in stochastic approximation (SA), which encompasses a bunch of applications in optimization and machine learning. To that end, we analyze the almost-sure convergence rates of the averaged iterates to a scaled sum of Gaussians in both linear and nonlinear SA problems. We then construct three types of asymptotic confidence sequences that are valid uniformly across all times with coverage guarantees, in an asymptotic sense that the starting time is sufficiently large. These coverage guarantees remain valid if the unknown covariance matrix is replaced by its plug-in estimator, and we conduct experiments to validate our methodology.