# On the sample complexity of parameter estimation in logistic regression with normal design

arXiv: https://arxiv.org/abs/2307.04191 · HTML: https://mlsanity.com/paper/2307.04191/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 30 references that are in the map.
- First posted on arXiv 9 Jul 2023.

## Builds on (most cited first, 6 of 6)

- [A modern maximum-likelihood theory for high-dimensional logistic regression](https://mlsanity.com/paper/1803.06964/) (2018)
- [Logistic Regression: The Importance of Being Improper](https://mlsanity.com/paper/1803.09349/) (2018)
- [The Impact of Regularization on High-dimensional Logistic Regression](https://mlsanity.com/paper/1906.03761/) (2019)
- [Logistic Regression: Tight Bounds for Stochastic and Online Optimization](https://mlsanity.com/paper/1405.3843/) (2014)
- [Finite-sample analysis of M-estimators using self-concordance](https://mlsanity.com/paper/1810.06838/) (2018)
- [An improper estimator with optimal excess risk in misspecified density estimation and logistic regression](https://mlsanity.com/paper/1912.10784/) (2019)

## Built on by (most cited first, 13 of 13)

- [Benefits of Early Stopping in Gradient Descent for Overparameterized Logistic Regression](https://mlsanity.com/paper/2502.13283/) (2025)
- [Capturing Individual Human Preferences with Reward Features](https://mlsanity.com/paper/2503.17338/) (2025)
- [Fast and Multiphase Rates for Nearest Neighbor Classifiers](https://mlsanity.com/paper/2308.08247/) (2023)
- [Dimension-free uniform concentration bound for logistic regression](https://mlsanity.com/paper/2405.18055/) (2024)
- [Exact Recovery of Sparse Binary Vectors from Generalized Linear Measurements](https://mlsanity.com/paper/2502.16008/) (2025)
- [On the Accuracy of Newton Step and Influence Function Data Attributions](https://mlsanity.com/paper/2512.12572/) (2025)
- [Hard labels sampled from sparse targets mislead rotation invariant algorithms](https://mlsanity.com/paper/2603.20967/) (2026)
- [Finite-Sample Performance of Gradient Descent in Logistic Regression with Gaussian Design](https://mlsanity.com/paper/2606.21683/) (2026)
- [The Nah Bandit: Modeling User Non-compliance in Recommendation Systems](https://mlsanity.com/paper/2408.07897/) (2024)
- [Near-Polynomially Competitive Active Logistic Regression](https://mlsanity.com/paper/2503.05981/) (2025)
- [Improved generalization bounds for binary linear classification via isoperimetry](https://mlsanity.com/paper/2505.16713/) (2025)
- [Estimation-Prediction Tradeoff in Causal Probabilistic Temporal Graphs](https://mlsanity.com/paper/2606.28225/) (2026)
- [Minimax Optimal Estimator and Improved Error Rate for the MLE in Logistic Regression with Gaussian Design](https://mlsanity.com/paper/2608.17260/) (2026)

## Abstract (arXiv)

The logistic regression model is one of the most popular data generation model in noisy binary classification problems. In this work, we study the sample complexity of estimating the parameters of the logistic regression model up to a given ℓ2 error, in terms of the dimension and the inverse temperature, with standard normal covariates. The inverse temperature controls the signal-to-noise ratio of the data generation process. While both generalization bounds and asymptotic performance of the maximum-likelihood estimator for logistic regression are well-studied, the non-asymptotic sample complexity that shows the dependence on error and the inverse temperature for parameter estimation is absent from previous analyses. We show that the sample complexity curve has two change-points in terms of the inverse temperature, clearly separating the low, moderate, and high temperature regimes.