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1. Implicit Regularization Leads to Benign Overfitting for Sparse Linear Regression

   Zhou, Mo; Ge, Rong (January 2023, International Conference on Machine Learning)

   In deep learning, often the training process nds an interpolator (a solution with 0 training loss), but the test loss is still low. This phenomenon, known as benign overfitting, is a major mystery that received a lot of recent attention. One common mechanism for benign overfitting is implicit regularization, where the training process leads to additional properties for the interpolator, often characterized by minimizing certain norms. However, even for a simple sparse linear regression problem y = Ax+ noise with sparse x , neither minimum l_1 orl_`2 norm interpolator gives the optimal test loss. In this work, we give a different parametrization of the model which leads to a new implicit regularization effect that combines the benefit of l_1 and l_2 interpolators. We show that training our new model via gradient descent leads to an interpolator with near-optimal test loss. Our result is based on careful analysis of the training dynamics and provides another example of implicit regularization effect that goes beyond norm minimization. 

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2. Deep learning: a statistical viewpoint

   https://doi.org/10.1017/S0962492921000027  

   Bartlett, Peter L.; Montanari, Andrea; Rakhlin, Alexander (May 2021, Acta Numerica)

   The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting, that is, accurate predictions despite overfitting training data. In this article, we survey recent progress in statistical learning theory that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behaviour of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favourable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings. 

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3. Federated Choquet Regression with LASSO for Outcome Prediction in Multisite Longitudinal Trial Data

   https://doi.org/10.1145/3761824  

   Lomasov, Semyon; Fang, Hua; Wang, Honggang (October 2025, ACM Transactions on Computing for Healthcare)

   Aggregating person-level data across multiple clinical study sites is often constrained by privacy regulations, necessitating the development of decentralized modeling approaches in biomedical research. To address this requirement, a federated nonlinear regression algorithm based on the Choquet integral has been introduced for outcome prediction. This approach avoids reliance on prior statistical assumptions about data distribution and captures feature interactions, reflecting the non-additive nature of biomedical data characteristics. This work represents the first theoretical application of Choquet integral regression to multisite longitudinal trial data within a federated learning framework. The Multiple Imputation Choquet Integral Regression with LASSO (MIChoquet-LASSO) algorithm is specifically designed to reduce overfitting and enable variable selection in federated learning settings. Its performance has been evaluated using synthetic datasets, publicly available biomedical datasets, and proprietary longitudinal randomized controlled trial data. Comparative evaluations were conducted against benchmark methods, including ordinary least squares (OLS) regression and Choquet-OLS regression, under various scenarios such as model misspecification and both linear and nonlinear data structures in non-federated and federated contexts. Mean squared error was used as the primary performance metric. Results indicate that MIChoquet-LASSO outperforms compared models in handling nonlinear longitudinal data with missing values, particularly in scenarios prone to overfitting. In federated settings, Choquet-OLS underperforms, whereas the federated variant of the model, FEDMIChoquet-LASSO, demonstrates consistently better performance. These findings suggest that FEDMIChoquet-LASSO offers a reliable solution for outcome prediction in multisite longitudinal trials, addressing challenges such as missing values, nonlinear relationships, and privacy constraints while maintaining strong performance within the federated learning framework. 

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4. Stable Minima Cannot Overfit in Univariate ReLU Networks: Generalization by Large Step Sizes

   Qiao, Dan; Zhang, Kaiqi; Singh, Esha; Soudry, Daniel; Wang, Yu-Xiang (October 2024, Advances in neural information processing systems)

   We study the generalization of two-layer ReLU neural networks in a univariate nonparametric regression problem with noisy labels. This is a problem where kernels (\emph{e.g.} NTK) are provably sub-optimal and benign overfitting does not happen, thus disqualifying existing theory for interpolating (0-loss, global optimal) solutions. We present a new theory of generalization for local minima that gradient descent with a constant learning rate can \emph{stably} converge to. We show that gradient descent with a fixed learning rate η can only find local minima that represent smooth functions with a certain weighted \emph{first order total variation} bounded by 1/η−1/2+O˜(σ+MSE‾‾‾‾‾√) where σ is the label noise level, MSE is short for mean squared error against the ground truth, and O˜(⋅) hides a logarithmic factor. Under mild assumptions, we also prove a nearly-optimal MSE bound of O˜(n−4/5) within the strict interior of the support of the n data points. Our theoretical results are validated by extensive simulation that demonstrates large learning rate training induces sparse linear spline fits. To the best of our knowledge, we are the first to obtain generalization bound via minima stability in the non-interpolation case and the first to show ReLU NNs without regularization can achieve near-optimal rates in nonparametric regression. 

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5. Benign Overfitting in Adversarial Training of Neural Networks

   Wang, Yunjuan; Zhang, Kaibo; Arora, Raman (July 2024, Proceedings of the 41st International Conference on Machine Learning, PMLR 235, 2024)

   Benign overfitting is the phenomenon wherein none of the predictors in the hypothesis class can achieve perfect accuracy (i.e., non-realizable or noisy setting), but a model that interpolates the training data still achieves good generalization. A series of recent works aim to understand this phenomenon for regression and classification tasks using linear predictors as well as two-layer neural networks. In this paper, we study such a benign overfitting phenomenon in an adversarial setting. We show that under a distributional assumption, interpolating neural networks found using adversarial training generalize well despite inferencetime attacks. Specifically, we provide convergence and generalization guarantees for adversarial training of two-layer networks (with smooth as well as non-smooth activation functions) showing that under moderate ℓ2 norm perturbation budget, the trained model has near-zero robust training loss and near-optimal robust generalization error. We support our theoretical findings with an empirical study on synthetic and real-world data. 

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