This paper proposes a new meta-learning method -- named HARMLESS (HAwkes Relational Meta LEarning method for Short Sequences) for learning heterogeneous point process models from short event sequence data along with a relational network. Specifically, we propose a hierarchical Bayesian mixture Hawkes process model, which naturally incorporates the relational information among sequences into point process modeling. Compared with existing methods, our model can capture the underlying mixed-community patterns of the relational network, which simultaneously encourages knowledge sharing among sequences and facilitates adaptive learning for each individual sequence. We further propose an efficient stochastic variational meta expectation maximization algorithm that can scale to large problems. Numerical experiments on both synthetic and real data show that HARMLESS outperforms existing methods in terms of predicting the future events.

The learning rate warmup heuristic achieves remarkable success in stabilizing training, accelerating convergence and improving generalization for adaptive stochastic optimization algorithms like RMSprop and Adam. Here, we study its mechanism in details. Pursuing the theory behind warmup, we identify a problem of the adaptive learning rate (i.e., it has problematically large variance in the early stage), suggest warmup works as a variance reduction technique, and provide both empirical and theoretical evidence to verify our hypothesis. We further propose RAdam, a new variant of Adam, by introducing a term to rectify the variance of the adaptive learning rate. Extensive experimental results on image classification, language modeling, and neural machine translation verify our intuition and demonstrate the effectiveness and robustness of our proposed method. All implementations are available at: https://github.com/LiyuanLucasLiu/RAdam.

Deep neural networks have revolutionized many real world applications, due to their flexibility in data fitting and accurate predictions for unseen data. A line of research reveals that neural networks can approximate certain classes of functions with an arbitrary accuracy, while the size of the network scales exponentially with respect to the data dimension. Empirical results, however, suggest that networks of moderate size already yield appealing performance. To explain such a gap, a common belief is that many data sets exhibit low dimensional structures, and can be modeled as samples near a low dimensional manifold. In this paper, we prove that neural networks can efficiently approximate functions supported on low dimensional manifolds. The network size scales exponentially in the approximation error, with an exponent depending on the intrinsic dimension of the data and the smoothness of the function. Our result shows that exploiting low dimensional data structures can greatly enhance the efficiency in function approximation by neural networks. We also implement a sub-network that assigns input data to their corresponding local neighborhoods, which may be of independent interest.

Optimal Transport (OT) naturally arises in many machine learning applications, yet the heavy computational burden limits its wide-spread uses. To address the scalability issue, we propose an implicit generative learning-based framework called SPOT (Scalable Push-forward of Optimal Transport). Specifically, we approximate the optimal transport plan by a pushforward of a reference distribution, and cast the optimal transport problem into a minimax problem. We then can solve OT problems efficiently using primal dual stochastic gradient-type algorithms. We also show that we can recover the density of the optimal transport plan using neural ordinary differential equations. Numerical experiments on both synthetic and real datasets illustrate that SPOT is robust and has favorable convergence behavior. SPOT also allows us to efficiently sample from the optimal transport plan, which benefits downstream applications such as domain adaptation.

Generative Adversarial Networks (GANs), though powerful, is hard to train. Several recent works (brock2016neural,miyato2018spectral) suggest that controlling the spectra of weight matrices in the discriminator can significantly improve the training of GANs. Motivated by their discovery, we propose a new framework for training GANs, which allows more flexible spectrum control (e.g., making the weight matrices of the discriminator have slow singular value decays). Specifically, we propose a new reparameterization approach for the weight matrices of the discriminator in GANs, which allows us to directly manipulate the spectra of the weight matrices through various regularizers and constraints, without intensively computing singular value decompositions. Theoretically, we further show that the spectrum control improves the generalization ability of GANs. Our experiments on CIFAR-10, STL-10, and ImageNet datasets confirm that compared to other methods, our proposed method is capable of generating images with competitive quality by utilizing spectral normalization and encouraging the slow singular value decay.

This decade has witnessed great breakthroughs in deep learning in a variety of applications, such as computer vision (Taigman et al., 2014; Girshick et al., 2014; He et al., 2016; Liu et al., 2017). Recent studies (Szegedy et al., 2013), however, show that most of these deep learning models are very vulnerable to adversarial attacks. Specifically, by injecting a small perturbation to the normal sample, attackers obtain the adversarial examples. Although these adversarial examples are semantically indistinguishable from the normal ones, they can severely fool the deep learning models and undermine the security of deep learning, causing reliability problems in autonomous driving, biometric authentication, etc. Researchers have devoted many effects to studying efficient adversarial attack and defense (Szegedy et al., 2013; Goodfellow et al., 2014b; Nguyen et al., 2015; Zheng et al., 2016; Madry et al., 2017). There is a growing body of work on generating successful adversarial examples, e.g., fast gradient sign method (FGSM, Goodfellow et al. (2014b)), projected gradient method (PGM, Kurakin et al. (2016)), etc. As for robustness, Goodfellow et al. (2014b) first propose to robustify the

Many machine learning techniques sacrifice convenient computational structures to gain estimation robustness and modeling flexibility. However, by exploring the modeling structures, we find these "sacrifices" do not always require more computational efforts. To shed light on such a "free-lunch" phenomenon, we study the square-root-Lasso (SQRT-Lasso) type regression problem. Specifically, we show that the nonsmooth loss functions of SQRT-Lasso type regression ease tuning effort and gain adaptivity to inhomogeneous noise, but is not necessarily more challenging than Lasso in computation. We can directly apply proximal algorithms (e.g. proximal gradient descent, proximal Newton, and proximal Quasi-Newton algorithms) without worrying the nonsmoothness of the loss function. Theoretically, we prove that the proximal algorithms combined with the pathwise optimization scheme enjoy fast convergence guarantees with high probability. Numerical results are provided to support our theory.