Transformers have demonstrated remarkable success across various applications. However, the success of transformers have not been understood in theory. In this work, we give a case study of how transformers can be trained to learn a classic statistical model with "group sparsity", where the input variables form multiple groups, and the label only depends on the variables from one of the groups. We theoretically demonstrate that, a one-layer transformer trained by gradient descent can correctly leverage the attention mechanism to select variables, disregarding irrelevant ones and focusing on those beneficial for classification. We also demonstrate that a well-pretrained one-layer transformer can be adapted to new downstream tasks to achieve good prediction accuracy with a limited number of samples. Our study sheds light on how transformers effectively learn structured data.

Gradient regularization, as described in Barrett and Dherin (in: International conference on learning representations, 2021), is a highly effective technique for promoting flat minima during gradient descent. Empirical evidence suggests that this regularization technique can significantly enhance the robustness of deep learning models against noisy perturbations, while also reducing test error. In this paper, we explore the per-example gradient regularization (PEGR) and present a theoretical analysis that demonstrates its effectiveness in improving both test error and robustness against noise perturbations. Specifically, we adopt a signal-noise data model from Cao et al. (Adv Neural Inf Process Syst 35:25237–25250, 2022) and show that PEGR can learn signals effectively while suppressing noise memorization. In contrast, standard gradient descent struggles to distinguish the signal from the noise, leading to suboptimal generalization performance. Our analysis reveals that PEGR penalizes the variance of pattern learning, thus effectively suppressing the memorization of noises from the training data. These findings underscore the importance of variance control in deep learning training and offer useful insights for developing more effective training approaches.

Modern deep learning models are usually highly over-parameterized so that they can overfit the training data. Surprisingly, such overfitting neural networks can usually still achieve high prediction accuracy. To study this “benign overfitting” phenomenon, a line of recent works has theoretically studied the learning of linear models and two-layer neural networks. However, most of these analyses are still limited to the very simple learning problems where the Bayes-optimal classifier is linear. In this work, we investigate a class of XOR-type classification tasks with label-flipping noises. We show that, under a certain condition on the sample complexity and signal-to-noise ratio, an over-parameterized ReLU CNN trained by gradient descent can achieve near Bayes-optimal accuracy. Moreover, we also establish a matching lower bound result showing that when the previous condition is not satisfied, the prediction accuracy of the obtained CNN is an absolute constant away from the Bayes-optimal rate. Our result demonstrates that CNNs have a remarkable capacity to efficiently learn XOR problems, even in the presence of highly correlated features.

Recent works have demonstrated a double descent phenomenon in over-parameterized learning. Although this phenomenon has been investigated by recent works, it has not been fully understood in theory. In this paper, we investigate the multiple descent phenomenon in a class of multi-component prediction models. We first consider a "double random feature model" (DRFM) concatenating two types of random features, and study the excess risk achieved by the DRFM in ridge regression. We calculate the precise limit of the excess risk under the high dimensional framework where the training sample size, the dimension of data, and the dimension of random features tend to infinity proportionally. Based on the calculation, we further theoretically demonstrate that the risk curves of DRFMs can exhibit triple descent. We then provide a thorough experimental study to verify our theory. At last, we extend our study to the "multiple random feature model" (MRFM), and show that MRFMs ensembling $K$ types of random features may exhibit $(K+1)$-fold descent. Our analysis points out that risk curves with a specific number of descent generally exist in learning multi-component prediction models.

Much recent research effort has been devoted to explain the success of deep learning. Random Matrix Theory (RMT) provides an emerging way to this end by analyzing the spectra of large random matrices involved in a trained deep neural network (DNN) such as weight matrices or Hessian matrices in the stochastic gradient descent algorithm. To better understand spectra of weight matrices, we conduct extensive experiments on weight matrices under different settings for layers, networks and data sets. Based on the previous work of {martin2018implicit}, spectra of weight matrices at the terminal stage of training are classified into three main types: Light Tail (LT), Bulk Transition period (BT) and Heavy Tail (HT). These different types, especially HT, implicitly indicate some regularization in the DNNs. In this paper, inspired from {martin2018implicit}, we identify the difficulty of the classification problem as an important factor for the appearance of HT in weight matrices spectra. Higher the classification difficulty, higher the chance for HT to appear. Moreover, the classification difficulty can be affected either by the signal-to-noise ratio of the dataset, or by the complexity of the classification problem (complex features, large number of classes) as well. Leveraging on this finding, we further propose a spectral criterion to detect the appearance of HT and use it to early stop the training process without testing data. Such early stopped DNNs have the merit of avoiding overfitting and unnecessary extra training while preserving a much comparable generalization ability. These findings from the paper are validated in several NNs (LeNet, MiniAlexNet and VGG), using Gaussian synthetic data and real data sets (MNIST and CIFAR10).