# Approximate Newton-based statistical inference using only stochastic gradients

arXiv: https://arxiv.org/abs/1805.08920 · HTML: https://mlsanity.com/paper/1805.08920/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2018.

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

- [Statistical Inference for Polyak-Ruppert Averaged Zeroth-order Stochastic Gradient Algorithm](https://mlsanity.com/paper/2102.05198/) (2021)
- [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)

## Abstract (arXiv)

We present a novel statistical inference framework for convex empirical risk minimization, using approximate stochastic Newton steps. The proposed algorithm is based on the notion of finite differences and allows the approximation of a Hessian-vector product from first-order information. In theory, our method efficiently computes the statistical error covariance in M-estimation, both for unregularized convex learning problems and high-dimensional LASSO regression, without using exact second order information, or resampling the entire data set. We also present a stochastic gradient sampling scheme for statistical inference in non-i.i.d. time series analysis, where we sample contiguous blocks of indices. In practice, we demonstrate the effectiveness of our framework on large-scale machine learning problems, that go even beyond convexity: as a highlight, our work can be used to detect certain adversarial attacks on neural networks.