# Online Statistical Inference for Stochastic Optimization via Kiefer-Wolfowitz Methods

arXiv: https://arxiv.org/abs/2102.03389 · HTML: https://mlsanity.com/paper/2102.03389/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 60 references that are in the map.
- First posted on arXiv 5 Feb 2021.

## Builds on (most cited first, 11 of 11)

- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [Optimal rates for zero-order convex optimization: the power of two function evaluations](https://mlsanity.com/paper/1312.2139/) (2013)
- [An Optimal Algorithm for Bandit and Zero-Order Convex Optimization with Two-Point Feedback](https://mlsanity.com/paper/1507.08752/) (2015)
- [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)
- [Asymptotic Optimality in Stochastic Optimization](https://mlsanity.com/paper/1612.05612/) (2016)
- [Online Covariance Matrix Estimation in Stochastic Gradient Descent](https://mlsanity.com/paper/2002.03979/) (2020)
- [Statistical Inference for Online Decision Making via Stochastic Gradient Descent](https://mlsanity.com/paper/2010.07341/) (2020)
- [Statistical Inference for the Population Landscape via Moment Adjusted Stochastic Gradients](https://mlsanity.com/paper/1712.07519/) (2017)
- [A generalization of regularized dual averaging and its dynamics](https://mlsanity.com/paper/1909.10072/) (2019)
- [On Scalable Inference with Stochastic Gradient Descent](https://mlsanity.com/paper/1707.00192/) (2017)

## Built on by (most cited first, 14 of 14)

- [Fast and Robust Online Inference with Stochastic Gradient Descent via Random Scaling](https://mlsanity.com/paper/2106.03156/) (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)
- [Online covariance estimation for stochastic gradient descent under Markovian sampling](https://mlsanity.com/paper/2308.01481/) (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)
- [Statistical Inference for Online Algorithms](https://mlsanity.com/paper/2505.17300/) (2025)
- [Adaptive debiased SGD in high-dimensional GLMs with streaming data](https://mlsanity.com/paper/2405.18284/) (2024)
- [On the SAGA algorithm with decreasing step](https://mlsanity.com/paper/2410.03760/) (2024)
- [Statistical Inference on Gradient Flows](https://mlsanity.com/paper/2606.01257/) (2026)
- [Enhancing Stochastic Optimization for Statistical Efficiency Using ROOT-SGD with Diminishing Stepsize](https://mlsanity.com/paper/2407.10955/) (2024)
- [A stochastic gradient descent algorithm with random search directions](https://mlsanity.com/paper/2503.19942/) (2025)
- [Online Statistical Inference of Constrained Stochastic Optimization via Random Scaling](https://mlsanity.com/paper/2505.18327/) (2025)

## Abstract (arXiv)

This paper investigates the problem of online statistical inference of model parameters in stochastic optimization problems via the Kiefer-Wolfowitz algorithm with random search directions. We first present the asymptotic distribution for the Polyak-Ruppert-averaging type Kiefer-Wolfowitz (AKW) estimators, whose asymptotic covariance matrices depend on the distribution of search directions and the function-value query complexity. The distributional result reflects the trade-off between statistical efficiency and function query complexity. We further analyze the choice of random search directions to minimize certain summary statistics of the asymptotic covariance matrix. Based on the asymptotic distribution, we conduct online statistical inference by providing two construction procedures of valid confidence intervals.