One of the fundamental problems in network analysis is detecting community structure in multi-layer networks, of which each layer represents one type of edge information among the nodes. We propose integrative spectral clustering approaches based on effective convex layer aggregations. Our aggregation methods are strongly motivated by a delicate asymptotic analysis of the spectral embedding of weighted adjacency matrices and the downstream $k$-means clustering, in a challenging regime where community detection consistency is impossible. In fact, the methods are shown to estimate the optimal convex aggregation, which minimizes the mis-clustering error under some specialized multi-layer network models. Our analysis further suggests that clustering using Gaussian mixture models is generally superior to the commonly used $k$-means in spectral clustering. Extensive numerical studies demonstrate that our adaptive aggregation techniques, together with Gaussian mixture model clustering, make the new spectral clustering remarkably competitive compared to several popularly used methods.

Graph-based semi-supervised node classification (GraphSSC) has wide applications, ranging from networking and security to data mining and machine learning, etc. However, existing centralized GraphSSC methods are impractical to solve many real-world graph-based problems, as collecting the entire graph and labeling a reasonable number of labels is time-consuming and costly, and data privacy may be also violated. Federated learning (FL) is an emerging learning paradigm that enables collaborative learning among multiple clients, which can mitigate the issue of label scarcity and protect data privacy as well. Therefore, performing GraphSSC under the FL setting is a promising solution to solve real-world graph-based problems. However, existing FL methods 1) perform poorly when data across clients are non-IID, 2) cannot handle data with new label domains, and 3) cannot leverage unlabeled data, while all these issues naturally happen in real-world graph-based problems. To address the above issues, we propose the first FL framework, namely GraphFL, for semi-supervised node classification on graphs. Our framework is motivated by meta-learning methods. Specifically, we propose two GraphFL methods to respectively address the non-IID issue in graph data and handle the tasks with new label domains. Furthermore, we design a self-training method to leverage unlabeled graph data. We adopt representative graph neural networks as GraphSSC methods and evaluate GraphFL on multiple graph datasets. Experimental results demonstrate that GraphFL significantly outperforms the compared FL baseline and GraphFL with self-training can obtain better performance.

The presence of spurious features interferes with the goal of obtaining robust models that perform well across many groups within the population. A natural remedy is to remove spurious features from the model. However, in this work we show that removal of spurious features can decrease accuracy due to the inductive biases of overparameterized models. We completely characterize how the removal of spurious features affects accuracy across different groups (more generally, test distributions) in noiseless overparameterized linear regression. In addition, we show that removal of spurious feature can decrease the accuracy even in balanced datasets -- each target co-occurs equally with each spurious feature; and it can inadvertently make the model more susceptible to other spurious features. Finally, we show that robust self-training can remove spurious features without affecting the overall accuracy. Experiments on the Toxic-Comment-Detectoin and CelebA datasets show that our results hold in non-linear models.

We show that standard machine-learning algorithms may be trained to predict certain invariants of low genus arithmetic curves. Using datasets of size around one hundred thousand, we demonstrate the utility of machine-learning in classification problems pertaining to the BSD invariants of an elliptic curve (including its rank and torsion subgroup), and the analogous invariants of a genus 2 curve. Our results show that a trained machine can efficiently classify curves according to these invariants with high accuracies (>0.97). For problems such as distinguishing between torsion orders, and the recognition of integral points, the accuracies can reach 0.998.

In this paper, a general stochastic optimization procedure is studied, unifying several variants of the stochastic gradient descent such as, among others, the stochastic heavy ball method, the Stochastic Nesterov Accelerated Gradient algorithm (S-NAG), and the widely used Adam algorithm. The algorithm is seen as a noisy Euler discretization of a non-autonomous ordinary differential equation, recently introduced by Belotto da Silva and Gazeau, which is analyzed in depth. Assuming that the objective function is non-convex and differentiable, the stability and the almost sure convergence of the iterates to the set of critical points are established. A noteworthy special case is the convergence proof of S-NAG in a non-convex setting. Under some assumptions, the convergence rate is provided under the form of a Central Limit Theorem. Finally, the non-convergence of the algorithm to undesired critical points, such as local maxima or saddle points, is established. Here, the main ingredient is a new avoidance of traps result for non-autonomous settings, which is of independent interest.

Many practical machine learning tasks can be framed as Structured prediction problems, where several output variables are predicted and considered interdependent. Recent theoretical advances in structured prediction have focused on obtaining fast rates convergence guarantees, especially in the Implicit Loss Embedding (ILE) framework. PAC-Bayes has gained interest recently for its capacity of producing tight risk bounds for predictor distributions. This work proposes a novel PAC-Bayes perspective on the ILE Structured prediction framework. We present two generalization bounds, on the risk and excess risk, which yield insights into the behavior of ILE predictors. Two learning algorithms are derived from these bounds.

Incorporating k-means-like clustering techniques into (deep) autoencoders constitutes an interesting idea as the clustering may exploit the learned similarities in the embedding to compute a non-linear grouping of data at-hand. Unfortunately, the resulting contributions are often limited by ad-hoc choices, decoupled optimization problems and other issues. We present a theoretically-driven deep clustering approach that does not suffer from these limitations and allows for joint optimization of clustering and embedding. The network in its simplest form is derived from a Gaussian mixture model and can be incorporated seamlessly into deep autoencoders for state-of-the-art performance.

Mixed-SCORE is a recent approach for mixed membership community detection proposed by Jin et al. (2017) which is an extension of SCORE (Jin, 2015). In the note Jin et al. (2018), the authors propose SCORE+ as an improvement of SCORE to handle with weak signal networks. In this paper, we propose a method called Mixed-SCORE+ designed based on the Mixed-SCORE and SCORE+, therefore Mixed-SCORE+ inherits nice properties of both Mixed-SCORE and SCORE+. In the proposed method, we consider K+1 eigenvectors when there are K communities to detect weak signal networks. And we also construct vertices hunting and membership reconstruction steps to solve the problem of mixed membership community detection. Compared with several benchmark methods, numerical results show that Mixed-SCORE+ provides a significant improvement on the Polblogs network and two weak signal networks Simmons and Caltech, with error rates 54/1222, 125/1137 and 94/590, respectively. Furthermore, Mixed-SCORE+ enjoys excellent performances on the SNAP ego-networks.

Modern neural network architectures typically have many millions of parameters and can be pruned significantly without substantial loss in effectiveness which demonstrates they are over-parameterized. The contribution of this work is two-fold. The first is a method for approximating a multivariate Bernoulli random variable by means of a deterministic and differentiable transformation of any real-valued multivariate random variable. The second is a method for model selection by element-wise multiplication of parameters with approximate binary gates that may be computed deterministically or stochastically and take on exact zero values. Sparsity is encouraged by the inclusion of a surrogate regularization to the $L_0$ loss. Since the method is differentiable it enables straightforward and efficient learning of model architectures by an empirical risk minimization procedure with stochastic gradient descent and theoretically enables conditional computation during training. The method also supports any arbitrary group sparsity over parameters or activations and therefore offers a framework for unstructured or flexible structured model pruning. To conclude experiments are performed to demonstrate the effectiveness of the proposed approach.

Currently the amount of data produced worldwide is increasing beyond measure, thus a high volume of unsupervised data must be processed continuously. One of the main unsupervised data analysis is clustering. In streaming data scenarios, the data is composed by an increasing sequence of batches of samples where the concept drift phenomenon may happen. In this paper, we formally define the Streaming $K$-means(S$K$M) problem, which implies a restart of the error function when a concept drift occurs. We propose a surrogate error function that does not rely on concept drift detection. We proof that the surrogate is a good approximation of the S$K$M error. Hence, we suggest an algorithm which minimizes this alternative error each time a new batch arrives. We present some initialization techniques for streaming data scenarios as well. Besides providing theoretical results, experiments demonstrate an improvement of the converged error for the non-trivial initialization methods.