Overparameterized models have recently challenged conventional learning theory by exhibiting improved generalization beyond the interpolation limit, a phenomenon known as benign overfitting. This work introduces Adaptive Benign Overfitting (ABO), extending the recursive least-squares (RLS) framework to this regime through a numerically stable formulation based on orthogonal-triangular updates. A QR-based exponentially weighted RLS (QR-EWRLS) algorithm is introduced, combining random Fourier feature mappings with forgetting-factor regularization to enable online adaptation under non-stationary conditions. The orthogonal decomposition prevents the numerical divergence associated with covariance-form RLS while retaining adaptability to evolving data distributions. Experiments on nonlinear synthetic time series confirm that the proposed approach maintains bounded residuals and stable condition numbers while reproducing the double-descent behavior characteristic of overparameterized models. Applications to forecasting foreign exchange and electricity demand show that ABO is highly accurate (comparable to baseline kernel methods) while achieving speed improvements of between 20 and 40 percent. The results provide a unified view linking adaptive filtering, kernel approximation, and benign overfitting within a stable online learning framework.

Large neural networks trained in the overparameterized regime are able to fit noise to zero train error. Recent work of Nakkiran and Bansal has empirically observed that such networks behave as "conditional samplers" from the noisy distribution. That is, they replicate the noise in the train data to unseen examples. We give a theoretical framework for studying this conditional sampling behavior in the context of learning theory. We relate the notion of such samplers to knowledge distillation, where a student network imitates the outputs of a teacher on unlabeled data. We show that samplers, while being bad classifiers, can be good teachers. Concretely, we prove that distillation from samplers is guaranteed to produce a student which approximates the Bayes optimal classifier. Finally, we show that some common learning algorithms (e.g., Nearest-Neighbours and Kernel Machines) can often generate samplers when applied in the overparameterized regime.

Our general setup leads to a number of interesting findings. We outline precise conditions that decide the sign of the optimal setting $\lambda_{\opt}$ for the ridge parameter $\lambda$ and confirm the implicit $\ell_2$ regularization effect of overparameterization, which theoretically justifies the surprising empirical observation that $\lambda_{\opt}$ can be \textit{negative} in the overparameterized regime. We also characterize the double descent phenomenon for principal component regression (PCR) when $\vX$ and $\vbeta_{\star}$ are both anisotropic. Finally, we determine the optimal weighting matrix $\vSigma_w$ for both the ridgeless ($\lambda\to 0$) and optimally regularized ($\lambda = \lambda_{\opt}$) case, and demonstrate the advantage of the weighted objective over standard ridge regression and PCR.

This paper studies the role of activation functions in learning modular addition with two-layer neural networks. We first establish a sharp expressivity gap: sine MLPs admit width-$2$ exact realizations for any fixed length $m$ and, with bias, width-$2$ exact realizations uniformly over all lengths. In contrast, the width of ReLU networks must scale linearly with $m$ to interpolate, and they cannot simultaneously fit two lengths with different residues modulo $p$. We then provide a novel Natarajan-dimension generalization bound for sine networks, yielding nearly optimal sample complexity $\widetilde{\mathcal{O}}(p)$ for ERM over constant-width sine networks. We also derive width-independent, margin-based generalization for sine networks in the overparametrized regime and validate it. Empirically, sine networks generalize consistently better than ReLU networks across regimes and exhibit strong length extrapolation.

Global convergence and induced kernels of gradient-based meta-learning with neural nets.

In supervised classification tasks, models are trained to predict a label for each data point. In real-world datasets, these labels are often noisy due to annotation errors. While the impact of label noise on the performance of deep learning models has been widely studied, its effects on the networks' hidden representations remain poorly understood. We address this gap by systematically comparing hidden representations using the Information Imbalance, a computationally efficient proxy of conditional mutual information. Through this analysis, we observe that the information content of the hidden representations follows a double descent as a function of the number of network parameters, akin to the behavior of the test error. We further demonstrate that in the underparameterized regime, representations learned with noisy labels are more informative than those learned with clean labels, while in the overparameterized regime, these representations are equally informative. Our results indicate that the representations of overparameterized networks are robust to label noise. We also found that the information imbalance between the penultimate and pre-softmax layers decreases with cross-entropy loss in the overparameterized regime. This offers a new perspective on understanding generalization in classification tasks. Extending our analysis to representations learned from random labels, we show that these perform worse than random features. This indicates that training on random labels drives networks much beyond lazy learning, as weights adapt to encode labels information.

Data scarcity drives the need for more sample-efficient large language models. In this work, we use the double descent phenomenon to holistically compare the sample efficiency of discrete diffusion and autoregressive models. We show that discrete diffusion models require larger capacity and more training epochs to escape their underparameterized regime and reach the interpolation threshold. In the strongly overparameterized regime, both models exhibit similar behavior, with neither exhibiting a pronounced second descent in test loss across a large range of model sizes. Overall, our results indicate that autoregressive models are more sample-efficient on small-scale datasets, while discrete diffusion models only become competitive when given sufficient capacity and compute.