The success of boosted classifiers is most often attributed to improvements in margins.

Recent works on over-parameterized neural networks have shown that the stochasticity in optimizers has the implicit regularization effect of minimizing the sharpness of the loss function (in particular, the trace of its Hessian) over the family zero-loss solutions. More explicit forms of flatness regularization also empirically improve the generalization performance. However, it remains unclear why and when flatness regularization leads to better generalization. This work takes the first step towards understanding the inductive bias of the minimum trace of the Hessian solutions in an important setting: learning deep linear networks from linear measurements, also known as \emph{deep matrix factorization}. We show that with the standard Restricted Isometry Property (RIP) on the measurements, minimizing the trace of Hessian is approximately equivalent to minimizing the Schatten 1-norm of the corresponding end-to-end matrix parameters (i.e., the product of all layer matrices), which in turn leads to better generalization.

Recently, there is a growing interest in studying pairwise learning since it includes many important machine learning tasks as specific examples, e.g., metric learning, AUC maximization and ranking. While stochastic gradient descent (SGD) is an efficient method, there is a lacking study on its generalization behavior for pairwise learning. In this paper, we present a systematic study on the generalization analysis of SGD for pairwise learning to understand the balance between generalization and optimization. We develop a novel high-probability generalization bound for uniformly-stable algorithms to incorporate the variance information for better generalization, based on which we establish the first nonsmooth learning algorithm to achieve almost optimal high-probability and dimension-independent generalization bounds in linear time. We consider both convex and nonconvex pairwise learning problems. Our stability analysis for convex problems shows how the interpolation can help generalization. We establish a uniform convergence of gradients, and apply it to derive the first generalization bounds on population gradients for nonconvex problems. Finally, we develop better generalization bounds for gradient-dominated problems.

Regularization can mitigate the generalization gap between training and inference by introducing inductive bias. Existing works have already proposed various inductive biases from diverse perspectives. However, none of them explores inductive bias from the perspective of class-dependent response distribution of individual neurons. In this paper, we conduct a substantial analysis of the characteristics of such distribution. Based on the analysis results, we articulate the Neuron Steadiness Hypothesis: the neuron with similar responses to instances of the same class leads to better generalization. Accordingly, we propose a new regularization method called Neuron Steadiness Regularization (NSR) to reduce neuron intra-class response variance. Based on the Complexity Measure, we theoretically guarantee the effectiveness of NSR for improving generalization. We conduct extensive experiments on Multilayer Perceptron, Convolutional Neural Networks, and Graph Neural Networks with popular benchmark datasets of diverse domains, which show that our Neuron Steadiness Regularization consistently outperforms the vanilla version of models with significant gain and low additional computational overhead.

Machine learning models that are developed to be invariant under certain types of data transformations have shown improved generalization in practice. However, a principled understanding of why invariance benefits generalization is limited. Given a dataset, there is often no principled way to select suitable data transformations under which model invariance guarantees better generalization. This paper studies the generalization benefit of model invariance by introducing the sample cover induced by transformations, i.e., a representative subset of a dataset that can approximately recover the whole dataset using transformations. For any data transformations, we provide refined generalization bounds for invariant models based on the sample cover. We also characterize the suitability of a set of data transformations by the sample covering number induced by transformations, i.e., the smallest size of its induced sample covers. We show that we may tighten the generalization bounds for suitable transformations that have a small sample covering number. In addition, our proposed sample covering number can be empirically evaluated and thus provides a guidance for selecting transformations to develop model invariance for better generalization. In experiments on multiple datasets, we evaluate sample covering numbers for some commonly used transformations and show that the smaller sample covering number for a set of transformations (e.g., the 3D-view transformation) indicates a smaller gap between the test and training error for invariant models, which verifies our propositions.

Adaptive gradient methods such as AdaGrad, RMSprop and Adam have been optimizers of choice for deep learning due to their fast training speed. However, it was recently observed that their generalization performance is often worse than that of SGD for over-parameterized neural networks. While new algorithms such as AdaBound, SWAT, and Padam were proposed to improve the situation, the provided analyses are only committed to optimization bounds for the training objective, leaving critical generalization capacity unexplored. To close this gap, we propose \textit{\textbf{S}table \textbf{A}daptive \textbf{G}radient \textbf{D}escent} (\textsc{SAGD}) for nonconvex optimization which leverages differential privacy to boost the generalization performance of adaptive gradient methods. Theoretical analyses show that \textsc{SAGD} has high-probability convergence to a population stationary point.

Despite extensive studies, the underlying reason as to why overparameterizedneural networks can generalize remains elusive. Existing theory shows that common stochastic optimizers prefer flatter minimizers of the training loss, and thusa natural potential explanation is that flatness implies generalization. This workcritically examines this explanation. Through theoretical and empirical investigation, we identify the following three scenarios for two-layer ReLU networks: (1)flatness provably implies generalization; (2) there exist non-generalizing flattestmodels and sharpness minimization algorithms fail to generalize poorly, and (3)perhaps most strikingly, there exist non-generalizing flattest models, but sharpnessminimization algorithms still generalize. Our results suggest that the relationshipbetween sharpness and generalization subtly depends on the data distributionsand the model architectures and sharpness minimization algorithms do not onlyminimize sharpness to achieve better generalization.

For heterophilic graphs, achieving balance based on relative gradients leads to faster training and better generalization. In contrast, homophilic graphs benefit from delaying the learning of later layers.

We use a sliding window approach to generate samples of sequences. We forecast in an autoregressive manner to generate multi-step ahead predictions. We compare our model with a series of baselines on the multi-step forecasting with different dynamics. Apart from the root mean square error (RMSE), we also report the energy spectrum error (ESE) for ocean current prediction which quantifies the physical consistency. Our model achieves this by learning on multiple tasks simultaneously and then adapting to new tasks with domain transfer.