# An Analysis of Constant Step Size SGD in the Non-convex Regime: Asymptotic Normality and Bias

arXiv: https://arxiv.org/abs/2006.07904 · HTML: https://mlsanity.com/paper/2006.07904/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jun 2020.

## Built on by (most cited first, 30 of 48)

- [Manipulating SGD with Data Ordering Attacks](https://mlsanity.com/paper/2104.09667/) (2021)
- [Neural Networks Efficiently Learn Low-Dimensional Representations with SGD](https://mlsanity.com/paper/2209.14863/) (2022)
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- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Bias and Extrapolation in Markovian Linear Stochastic Approximation with Constant Stepsizes](https://mlsanity.com/paper/2210.00953/) (2022)
- [Fractal Structure and Generalization Properties of Stochastic Optimization Algorithms](https://mlsanity.com/paper/2106.04881/) (2021)
- [Constant Stepsize Q-learning: Distributional Convergence, Bias and Extrapolation](https://mlsanity.com/paper/2401.13884/) (2024)
- [Stationary Behavior of Constant Stepsize SGD Type Algorithms: An Asymptotic Characterization](https://mlsanity.com/paper/2111.06328/) (2021)
- [Bootstrapping the error of Oja's algorithm](https://mlsanity.com/paper/2106.14857/) (2021)
- [Generalization Bounds for Stochastic Gradient Descent via Localized varepsilon-Covers](https://mlsanity.com/paper/2209.08951/) (2022)
- [Convergence and concentration properties of constant step-size SGD through Markov chains](https://mlsanity.com/paper/2306.11497/) (2023)
- [High Confidence Level Inference is Almost Free using Parallel Stochastic Optimization](https://mlsanity.com/paper/2401.09346/) (2024)
- [Computing the Bias of Constant-step Stochastic Approximation with Markovian Noise](https://mlsanity.com/paper/2405.14285/) (2024)
- [Weighted Averaged Stochastic Gradient Descent: Asymptotic Normality and Optimality](https://mlsanity.com/paper/2307.06915/) (2023)
- [Stochastic Gradient Line Bayesian Optimization for Efficient Noise-Robust Optimization of Parameterized Quantum Circuits](https://mlsanity.com/paper/2111.07952/) (2021)
- [Online covariance estimation for stochastic gradient descent under Markovian sampling](https://mlsanity.com/paper/2308.01481/) (2023)
- [Statistical Inference for Polyak-Ruppert Averaged Zeroth-order Stochastic Gradient Algorithm](https://mlsanity.com/paper/2102.05198/) (2021)
- [High-dimensional scaling limits and fluctuations of online least-squares SGD with smooth covariance](https://mlsanity.com/paper/2304.00707/) (2023)
- [Stochastic Methods in Variational Inequalities: Ergodicity, Bias and Refinements](https://mlsanity.com/paper/2306.16502/) (2023)
- [Nonasymptotic Analysis of Stochastic Gradient Descent with the Richardson-Romberg Extrapolation](https://mlsanity.com/paper/2410.05106/) (2024)
- [Effectiveness of Constant Stepsize in Markovian LSA and Statistical Inference](https://mlsanity.com/paper/2312.10894/) (2023)
- [Two-Timescale Linear Stochastic Approximation: Constant Stepsizes Go a Long Way](https://mlsanity.com/paper/2410.13067/) (2024)
- [Robust Stochastic Optimization via Gradient Quantile Clipping](https://mlsanity.com/paper/2309.17316/) (2023)
- [Prelimit Coupling and Steady-State Convergence of Constant-stepsize Nonsmooth Contractive SA](https://mlsanity.com/paper/2404.06023/) (2024)
- [A Piecewise Lyapunov Analysis of Sub-quadratic SGD: Applications to Robust and Quantile Regression](https://mlsanity.com/paper/2504.08178/) (2025)
- [Beyond Sin-Squared Error: Linear-Time Entrywise Uncertainty Quantification for Streaming PCA](https://mlsanity.com/paper/2506.12655/) (2025)
- [On Uniform Boundedness Properties of SGD and its Momentum Variants](https://mlsanity.com/paper/2201.10245/) (2022)
- [The Effect of SGD Batch Size on Autoencoder Learning: Sparsity, Sharpness, and Feature Learning](https://mlsanity.com/paper/2308.03215/) (2023)
- [Demystifying the Myths and Legends of Nonconvex Convergence of SGD](https://mlsanity.com/paper/2310.12969/) (2023)
- [On the Rate of Gaussian Approximation for Linear Regression Problems](https://mlsanity.com/paper/2509.14039/) (2025)

## Abstract (arXiv)

Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for such problems. However, quantifying the uncertainty associated with the underlying training algorithm is not well-studied in the non-convex setting. In order to address this shortcoming, in this work, we establish an asymptotic normality result for the constant step size stochastic gradient descent (SGD) algorithm--a widely used algorithm in practice. Specifically, based on the relationship between SGD and Markov Chains [DDB19], we show that the average of SGD iterates is asymptotically normally distributed around the expected value of their unique invariant distribution, as long as the non-convex and non-smooth objective function satisfies a dissipativity property. We also characterize the bias between this expected value and the critical points of the objective function under various local regularity conditions. Together, the above two results could be leveraged to construct confidence intervals for non-convex problems that are trained using the SGD algorithm.