Extension to non-Gaussian designs

Determine the class of non-Gaussian covariate distributions for which the minimax-optimal estimation rates and efficient estimators established for logistic regression with Gaussian design remain valid, and extend the accompanying finite-sample arguments to those designs.

Background

The paper’s results rely substantially on Gaussian rotational invariance and Gaussian moment estimates. The authors note that a prior upper bound for the MLE extends, up to logarithmic factors, to a class of regular designs, but the paper does not characterize the broader design classes supporting its minimax lower bound, debiased norm estimator, or full parameter-estimation guarantees.

The unresolved task therefore has two components: identify an appropriate larger class of covariate distributions and establish extensions of the current proof techniques and results to that class.

References

Moreover, it is also interesting to investigate to what extent our results extend to non-Gaussian designs. In fact, it was shown that the upper bound of [3] remains valid, up to some logarithmic factors, over a class of regular designs; see [3, Theorem 3]. We expect that our results also extend to a larger class of designs, while it remains rather subtle to identify these designs and extend some of our current arguments.

— Minimax Optimal Estimator and Improved Error Rate for the MLE in Logistic Regression with Gaussian Design  (2608.17260 - Chen et al., 18 Aug 2026) in Section 7, Concluding Remarks