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1. Multiplier bootstrap for quantile regression: non-asymptotic theory under random design

   https://doi.org/10.1093/imaiai/iaaa006  

   Pan, Xiaoou; Zhou, Wen-Xin (May 2020, Information and Inference: A Journal of the IMA)

   null (Ed.)

   Abstract This paper establishes non-asymptotic concentration bound and Bahadur representation for the quantile regression estimator and its multiplier bootstrap counterpart in the random design setting. The non-asymptotic analysis keeps track of the impact of the parameter dimension $$d$$ and sample size $$n$$ in the rate of convergence, as well as in normal and bootstrap approximation errors. These results represent a useful complement to the asymptotic results under fixed design and provide theoretical guarantees for the validity of Rademacher multiplier bootstrap in the problems of confidence construction and goodness-of-fit testing. Numerical studies lend strong support to our theory and highlight the effectiveness of Rademacher bootstrap in terms of accuracy, reliability and computational efficiency. 

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2. Adaptive Linear Estimating Equations

   Ying, Mufang; Khamaru, Koulik; Zhang, Cun-Hui (May 2024, 2023 Conference on Neural Information Processing Systems)

   Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure. For instance, the ordinary least squares (OLS) estimator in an adaptive linear regression model can exhibit non-normal asymptotic behavior, posing challenges for accurate inference and interpretation. In this paper, we propose a general method for constructing debiased estimator which remedies this issue. It makes use of the idea of adaptive linear estimating equations, and we establish theoretical guarantees of asymptotic normality, supplemented by discussions on achieving near-optimal asymptotic variance. A salient feature of our estimator is that in the context of multi-armed bandits, our estimator retains the non-asymptotic performance of the least squares estimator while obtaining asymptotic normality property. Consequently, this work helps connect two fruitful paradigms of adaptive inference: a) non-asymptotic inference using concentration inequalities and b) asymptotic inference via asymptotic normality. 

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3. Adaptive linear estimating equations

   Ying, Mufang; Khamaru, Koulik; Zhang, Cun-Hui (May 2024, 37th Conference on Neural Information Processing Systems)

   Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure. For instance, the ordinary least squares (OLS) estimator in an adaptive linear regression model can exhibit non-normal asymptotic behavior, posing challenges for accurate inference and interpretation. In this paper, we propose a general method for constructing debiased estimator which remedies this issue. It makes use of the idea of adaptive linear estimating equations, and we establish theoretical guarantees of asymptotic normality, supplemented by discussions on achieving near-optimal asymptotic variance. A salient feature of our estimator is that in the context of multi-armed bandits, our estimator retains the non-asymptotic performance of the least square estimator while obtaining asymptotic normality property. Consequently, this work helps connect two fruitful paradigms of adaptive inference: a) non-asymptotic inference using concentration inequalities and b) asymptotic inference via asymptotic normality. 

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4. Overparameterized Random Feature Regression with Nearly Orthogonal Data

   Wang, Zhichao; Zhu, Yizhe (April 2023, Proceedings of Machine Learning Research)

   Ruiz, Francisco; Dy, Jennifer; van_de_Meent, Jan-Willem (Ed.)

   We investigate the properties of random feature ridge regression (RFRR) given by a two-layer neural network with random Gaussian initialization. We study the non-asymptotic behaviors of the RFRR with nearly orthogonal deterministic unit-length input data vectors in the overparameterized regime, where the width of the first layer is much larger than the sample size. Our analysis shows high-probability non-asymptotic concentration results for the training errors, cross-validations, and generalization errors of RFRR centered around their respective values for a kernel ridge regression (KRR). This KRR is derived from an expected kernel generated by a nonlinear random feature map. We then approximate the performance of the KRR by a polynomial kernel matrix obtained from the Hermite polynomial expansion of the activation function, whose degree only depends on the orthogonality among different data points. This polynomial kernel determines the asymptotic behavior of the RFRR and the KRR. Our results hold for a wide variety of activation functions and input data sets that exhibit nearly orthogonal properties. Based on these approximations, we obtain a lower bound for the generalization error of the RFRR for a nonlinear student-teacher model. 

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5. Statistical Optimality and Stability of Tangent Transform Algorithms in Logit Models

   Ghosh, I. (July 2022, Journal of machine learning research)

   Pierre Alquier (Ed.)

   A systematic approach to finding variational approximation in an otherwise intractable non-conjugate model is to exploit the general principle of convex duality by minorizing the marginal likelihood that renders the problem tractable. While such approaches are popular in the context of variational inference in non-conjugate Bayesian models, theoretical guarantees on statistical optimality and algorithmic convergence are lacking. Focusing on logistic regression models, we provide mild conditions on the data generating process to derive non-asymptotic upper bounds to the risk incurred by the variational optima. We demonstrate that these assumptions can be completely relaxed if one considers a slight variation of the algorithm by raising the likelihood to a fractional power. Next, we utilize the theory of dynamical systems to provide convergence guarantees for such algorithms in logistic and multinomial logit regression. In particular, we establish local asymptotic stability of the algorithm without any assumptions on the data-generating process. We explore a special case involving a semi-orthogonal design under which a global convergence is obtained. The theory is further illustrated using several numerical studies. 

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