# Statistical Inference for Polyak-Ruppert Averaged Zeroth-order Stochastic Gradient Algorithm arXiv: https://arxiv.org/abs/2102.05198 · HTML: https://mlsanity.com/paper/2102.05198/ ## Key facts - Cited by 6 papers in the map (data through 14 Oct 2026). - Builds on 36 of its 95 references that are in the map. - First posted on arXiv 10 Feb 2021. ## Builds on (most cited first, 30 of 36) - [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013) - [ZOO: Zeroth Order Optimization based Black-box Attacks to Deep Neural Networks without Training Substitute Models](https://mlsanity.com/paper/1708.03999/) (2017) - [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017) - [Black-box Adversarial Attacks with Limited Queries and Information](https://mlsanity.com/paper/1804.08598/) (2018) - [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018) - [Optimal rates for zero-order convex optimization: the power of two function evaluations](https://mlsanity.com/paper/1312.2139/) (2013) - [Efficient Decision-based Black-box Adversarial Attacks on Face Recognition](https://mlsanity.com/paper/1904.04433/) (2019) - [A Primer on Zeroth-Order Optimization in Signal Processing and Machine Learning](https://mlsanity.com/paper/2006.06224/) (2020) - [Query-Efficient Hard-label Black-box Attack:An Optimization-based Approach](https://mlsanity.com/paper/1807.04457/) (2018) - [Sign-OPT: A Query-Efficient Hard-label Adversarial Attack](https://mlsanity.com/paper/1909.10773/) (2019) - [An Optimal Algorithm for Bandit and Zero-Order Convex Optimization with Two-Point Feedback](https://mlsanity.com/paper/1507.08752/) (2015) - [Zeroth-Order Stochastic Variance Reduction for Nonconvex Optimization](https://mlsanity.com/paper/1805.10367/) (2018) - [NATTACK: Learning the Distributions of Adversarial Examples for an Improved Black-Box Attack on Deep Neural Networks](https://mlsanity.com/paper/1905.00441/) (2019) - [Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains](https://mlsanity.com/paper/1707.06386/) (2017) - [Risk and parameter convergence of logistic regression](https://mlsanity.com/paper/1803.07300/) (2018) - [ZO-AdaMM: Zeroth-Order Adaptive Momentum Method for Black-Box Optimization](https://mlsanity.com/paper/1910.06513/) (2019) - [Statistical Inference for Model Parameters in Stochastic Gradient Descent](https://mlsanity.com/paper/1610.08637/) (2016) - [Stochastic Zeroth-order Optimization in High Dimensions](https://mlsanity.com/paper/1710.10551/) (2017) - [Stochastic (Approximate) Proximal Point Methods: Convergence, Optimality, and Adaptivity](https://mlsanity.com/paper/1810.05633/) (2018) - [An Analysis of Constant Step Size SGD in the Non-convex Regime: Asymptotic Normality and Bias](https://mlsanity.com/paper/2006.07904/) (2020) - [An ADMM Based Framework for AutoML Pipeline Configuration](https://mlsanity.com/paper/1905.00424/) (2019) - [Improved Zeroth-Order Variance Reduced Algorithms and Analysis for Nonconvex Optimization](https://mlsanity.com/paper/1910.12166/) (2019) - [Zeroth-Order Regularized Optimization (ZORO): Approximately Sparse Gradients and Adaptive Sampling](https://mlsanity.com/paper/2003.13001/) (2020) - [A Comprehensive Linear Speedup Analysis for Asynchronous Stochastic Parallel Optimization from Zeroth-Order to First-Order](https://mlsanity.com/paper/1606.00498/) (2016) - [Gradientless Descent: High-Dimensional Zeroth-Order Optimization](https://mlsanity.com/paper/1911.06317/) (2019) - [Asymptotic Optimality in Stochastic Optimization](https://mlsanity.com/paper/1612.05612/) (2016) - [Zeroth-order Nonconvex Stochastic Optimization: Handling Constraints, High-Dimensionality and Saddle-Points](https://mlsanity.com/paper/1809.06474/) (2018) - [Normal Approximation for Stochastic Gradient Descent via Non-Asymptotic Rates of Martingale CLT](https://mlsanity.com/paper/1904.02130/) (2019) - [A Frank-Wolfe Framework for Efficient and Effective Adversarial Attacks](https://mlsanity.com/paper/1811.10828/) (2018) - [Zeroth-Order Online Alternating Direction Method of Multipliers: Convergence Analysis and Applications](https://mlsanity.com/paper/1710.07804/) (2017) ## Built on by (most cited first, 6 of 6) - [Online covariance estimation for stochastic gradient descent under Markovian sampling](https://mlsanity.com/paper/2308.01481/) (2023) - [Statistical Inference for Linear Functionals of Online SGD in High-dimensional Linear Regression](https://mlsanity.com/paper/2302.09727/) (2023) - [Hebbian learning inspired estimation of the linear regression parameters from queries](https://mlsanity.com/paper/2311.03483/) (2023) - [Statistical Inference for Linear Functionals of Online Least-squares SGD when t gtrsim d1+δ](https://mlsanity.com/paper/2510.19734/) (2025) - [Enhancing Stochastic Optimization for Statistical Efficiency Using ROOT-SGD with Diminishing Stepsize](https://mlsanity.com/paper/2407.10955/) (2024) - [Direct Fisher Score Estimation for Likelihood Maximization](https://mlsanity.com/paper/2506.06542/) (2025) ## Abstract (arXiv) Statistical machine learning models trained with stochastic gradient algorithms are increasingly being deployed in critical scientific applications. However, computing the stochastic gradient in several such applications is highly expensive or even impossible at times. In such cases, derivative-free or zeroth-order algorithms are used. An important question which has thus far not been addressed sufficiently in the statistical machine learning literature is that of equipping stochastic zeroth-order algorithms with practical yet rigorous inferential capabilities so that we not only have point estimates or predictions but also quantify the associated uncertainty via confidence intervals or sets. Towards this, in this work, we first establish a central limit theorem for Polyak-Ruppert averaged stochastic zeroth-order gradient algorithm. We then provide online estimators of the asymptotic covariance matrix appearing in the central limit theorem, thereby providing a practical procedure for constructing asymptotically valid confidence sets (or intervals) for parameter estimation (or prediction) in the zeroth-order setting.