# Fast and Robust Online Inference with Stochastic Gradient Descent via Random Scaling

arXiv: https://arxiv.org/abs/2106.03156 · HTML: https://mlsanity.com/paper/2106.03156/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 36 references that are in the map.
- First posted on arXiv 6 Jun 2021.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)](https://mlsanity.com/paper/1306.2119/) (2013)
- [Statistical Inference for Model Parameters in Stochastic Gradient Descent](https://mlsanity.com/paper/1610.08637/) (2016)
- [On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration](https://mlsanity.com/paper/2004.04719/) (2020)
- [Online Covariance Matrix Estimation in Stochastic Gradient Descent](https://mlsanity.com/paper/2002.03979/) (2020)
- [Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT](https://mlsanity.com/paper/1904.02130/) (2019)
- [Online Statistical Inference for Stochastic Optimization via Kiefer-Wolfowitz Methods](https://mlsanity.com/paper/2102.03389/) (2021)
- [On Constructing Confidence Region for Model Parameters in Stochastic Gradient Descent via Batch Means](https://mlsanity.com/paper/1911.01483/) (2019)
- [Statistical Inference for the Population Landscape via Moment Adjusted Stochastic Gradients](https://mlsanity.com/paper/1712.07519/) (2017)

## Built on by (most cited first, 28 of 28)

- [A Statistical Analysis of Polyak-Ruppert Averaged Q-learning](https://mlsanity.com/paper/2112.14582/) (2021)
- [Online Statistical Inference for Nonlinear Stochastic Approximation with Markovian Data](https://mlsanity.com/paper/2302.07690/) (2023)
- [High Confidence Level Inference is Almost Free using Parallel Stochastic Optimization](https://mlsanity.com/paper/2401.09346/) (2024)
- [Online Statistical Inference for Contextual Bandits via Stochastic Gradient Descent](https://mlsanity.com/paper/2212.14883/) (2022)
- [Weighted Averaged Stochastic Gradient Descent: Asymptotic Normality and Optimality](https://mlsanity.com/paper/2307.06915/) (2023)
- [Statistical Inference with Stochastic Gradient Methods under φ-mixing Data](https://mlsanity.com/paper/2302.12717/) (2023)
- [Covariance Estimators for the ROOT-SGD Algorithm in Online Learning](https://mlsanity.com/paper/2212.01259/) (2022)
- [Effectiveness of Constant Stepsize in Markovian LSA and Statistical Inference](https://mlsanity.com/paper/2312.10894/) (2023)
- [SGMM: Stochastic Approximation to Generalized Method of Moments](https://mlsanity.com/paper/2308.13564/) (2023)
- [Asymptotic Time-Uniform Inference for Parameters in Averaged Stochastic Approximation](https://mlsanity.com/paper/2410.15057/) (2024)
- [Beyond Sin-Squared Error: Linear-Time Entrywise Uncertainty Quantification for Streaming PCA](https://mlsanity.com/paper/2506.12655/) (2025)
- [Statistical Estimation and Inference via Local SGD in Federated Learning](https://mlsanity.com/paper/2109.01326/) (2021)
- [Asymptotic Analysis of Sample-averaged Q-learning](https://mlsanity.com/paper/2410.10737/) (2024)
- [Statistical Inference for Online Algorithms](https://mlsanity.com/paper/2505.17300/) (2025)
- [Statistical Inference for Differentially Private Stochastic Gradient Descent](https://mlsanity.com/paper/2507.20560/) (2025)
- [Statistical Inference for Stochastic Gradient Descent: Beyond Finite Variance](https://mlsanity.com/paper/2605.26000/) (2026)
- [A Polyak-Ruppert Central Limit Theorem for SA-Adam with Momentum and Non-Convergent Adaptive Preconditioning](https://mlsanity.com/paper/2606.17364/) (2026)
- [How Much is Left? LLMs Linearly Encode Their Remaining Output Length](https://mlsanity.com/paper/2607.05316/) (2026)
- [Statistical Inference for Linear Functionals of Online SGD in High-dimensional Linear Regression](https://mlsanity.com/paper/2302.09727/) (2023)
- [Statistical Inference for Linear Functionals of Online Least-squares SGD when t gtrsim d1+δ](https://mlsanity.com/paper/2510.19734/) (2025)
- [Self-Normalized Inference for Constant-Stepsize Temporal-Difference Learning under Markovian Sampling](https://mlsanity.com/paper/2608.10896/) (2026)
- [Cheap Bootstrap for Fast Uncertainty Quantification of Stochastic Gradient Descent](https://mlsanity.com/paper/2310.11065/) (2023)
- [Enhancing Stochastic Optimization for Statistical Efficiency Using ROOT-SGD with Diminishing Stepsize](https://mlsanity.com/paper/2407.10955/) (2024)
- [Online Statistical Inference of Constrained Stochastic Optimization via Random Scaling](https://mlsanity.com/paper/2505.18327/) (2025)
- [Sharp asymptotic theory for Q-learning with LDTZ learning rate and its generalization](https://mlsanity.com/paper/2604.04218/) (2026)
- [Online Covariance Estimation in Averaged SGD: Improved Batch-Mean Rates and Minimax Optimality via Trajectory Regression](https://mlsanity.com/paper/2604.10814/) (2026)
- [Refining Covariance Matrix Estimation in Stochastic Gradient Descent Through Bias Reduction](https://mlsanity.com/paper/2604.21203/) (2026)
- [SAUSS: Stochastic Approximation with Unbiased Simulated Scores for Limited Dependent Variable Models](https://mlsanity.com/paper/2608.25304/) (2026)

## Abstract (arXiv)

We develop a new method of online inference for a vector of parameters estimated by the Polyak-Ruppert averaging procedure of stochastic gradient descent (SGD) algorithms. We leverage insights from time series regression in econometrics and construct asymptotically pivotal statistics via random scaling. Our approach is fully operational with online data and is rigorously underpinned by a functional central limit theorem. Our proposed inference method has a couple of key advantages over the existing methods. First, the test statistic is computed in an online fashion with only SGD iterates and the critical values can be obtained without any resampling methods, thereby allowing for efficient implementation suitable for massive online data. Second, there is no need to estimate the asymptotic variance and our inference method is shown to be robust to changes in the tuning parameters for SGD algorithms in simulation experiments with synthetic data.