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Multinomial Logistic Regression: Asymptotic Normality on Null Covariates in High-Dimensions
May 28, 2023
This paper investigates the asymptotic distribution of the maximum-likelihood estimate (MLE) in multinomial logistic models in the high-dimensional regime where dimension and sample size are of the same order. While classical large-sample theory provides asymptotic normality of the MLE under certain conditions, such classical results are expected to fail in high-dimensions as documented for the binary logistic case in the seminal work of Sur and Cand\`es [2019]. We address this issue in classification problems with 3 or more classes, by developing asymptotic normality and asymptotic chi-square results for the multinomial logistic MLE (also known as cross-entropy minimizer) on null covariates. Our theory leads to a new methodology to test the significance of a given feature. Extensive simulation studies on synthetic data corroborate these asymptotic results and confirm the validity of proposed p-values for testing the significance of a given feature.
De-Biasing The Lasso With Degrees-of-Freedom Adjustment
Feb 24, 2019This paper studies schemes to de-bias the Lasso in sparse linear regression where the goal is to estimate and construct confidence intervals for a low-dimensional projection of the unknown coefficient vector in a preconceived direction $a_0$. We assume that the design matrix has iid Gaussian rows with known covariance matrix $\Sigma$. Our analysis reveals that previous propositions to de-bias the Lasso require a modification in order to enjoy asymptotic efficiency in a full range of the level of sparsity. This modification takes the form of a degrees-of-freedom adjustment that accounts for the dimension of the model selected by the Lasso. Let $s_0$ denote the number of nonzero coefficients of the true coefficient vector. The unadjusted de-biasing schemes proposed in previous studies enjoys efficiency if $s_0\lll n^{2/3}$, up to logarithmic factors. However, if $s_0\ggg n^{2/3}$, the unadjusted scheme cannot be efficient in certain directions $a_0$. In the latter regime, it it necessary to modify existing procedures by an adjustment that accounts for the degrees-of-freedom of the Lasso. The proposed degrees-of-freedom adjustment grants asymptotic efficiency for any direction $a_0$. This holds under a Sparse Riecz Condition on the covariance matrix $\Sigma$ and the sample size requirement $s_0/p\to0$ and $s_0\log(p/s_0)/n\to0$. Our analysis also highlights that the degrees-of-freedom adjustment is not necessary when the initial bias of the Lasso in the direction $a_0$ is small, which is granted under more stringent conditions on $\Sigma^{-1}$. This explains why the necessity of degrees-of-freedom adjustment did not appear in some previous studies. The main proof argument involves a Gaussian interpolation path similar to that used to derive Slepian's lemma. It yields a sharp $\ell_\infty$ error bound for the Lasso under Gaussian design which is of independent interest.
On the prediction loss of the lasso in the partially labeled setting
Nov 08, 2016In this paper we revisit the risk bounds of the lasso estimator in the context of transductive and semi-supervised learning. In other terms, the setting under consideration is that of regression with random design under partial labeling. The main goal is to obtain user-friendly bounds on the off-sample prediction risk. To this end, the simple setting of bounded response variable and bounded (high-dimensional) covariates is considered. We propose some new adaptations of the lasso to these settings and establish oracle inequalities both in expectation and in deviation. These results provide non-asymptotic upper bounds on the risk that highlight the interplay between the bias due to the mis-specification of the linear model, the bias due to the approximate sparsity and the variance. They also demonstrate that the presence of a large number of unlabeled features may have significant positive impact in the situations where the restricted eigenvalue of the design matrix vanishes or is very small.