Machine learning develops rapidly, which has made many theoretical breakthroughs and is widely applied in various fields. Optimization, as an important part of machine learning, has attracted much attention of researchers. With the exponential growth of data amount and the increase of model complexity, optimization methods in machine learning face more and more challenges. A lot of work on solving optimization problems or improving optimization methods in machine learning has been proposed successively. The systematic retrospect and summary of the optimization methods from the perspective of machine learning are of great significance, which can offer guidance for both developments of optimization and machine learning research. In this paper, we first describe the optimization problems in machine learning. Then, we introduce the principles and progresses of commonly used optimization methods. Next, we summarize the applications and developments of optimization methods in some popular machine learning fields. Finally, we explore and give some challenges and open problems for the optimization in machine learning.

Social media sites are becoming a key factor in politics. These platforms are easy to manipulate for the purpose of distorting information space to confuse and distract voters. Past works to identify disruptive patterns are mostly focused on analyzing the content of tweets. In this study, we jointly embed the information from both user posted content as well as a user's follower network, to detect groups of densely connected users in an unsupervised fashion. We then investigate these dense sub-blocks of users to flag anomalous behavior. In our experiments, we study the tweets related to the upcoming 2019 Canadian Elections, and observe a set of densely-connected users engaging in local politics in different provinces, and exhibiting troll-like behavior.

Graph clustering is a fundamental task which discovers communities or groups in networks. Recent studies have mostly focused on developing deep learning approaches to learn a compact graph embedding, upon which classic clustering methods like k-means or spectral clustering algorithms are applied. These two-step frameworks are difficult to manipulate and usually lead to suboptimal performance, mainly because the graph embedding is not goal-directed, i.e., designed for the specific clustering task. In this paper, we propose a goal-directed deep learning approach, Deep Attentional Embedded Graph Clustering (DAEGC for short). Our method focuses on attributed graphs to sufficiently explore the two sides of information in graphs. By employing an attention network to capture the importance of the neighboring nodes to a target node, our DAEGC algorithm encodes the topological structure and node content in a graph to a compact representation, on which an inner product decoder is trained to reconstruct the graph structure. Furthermore, soft labels from the graph embedding itself are generated to supervise a self-training graph clustering process, which iteratively refines the clustering results. The self-training process is jointly learned and optimized with the graph embedding in a unified framework, to mutually benefit both components. Experimental results compared with state-of-the-art algorithms demonstrate the superiority of our method.

Graph-based clustering methods have demonstrated the effectiveness in various applications. Generally, existing graph-based clustering methods first construct a graph to represent the input data and then partition it to generate the clustering result. However, such a stepwise manner may make the constructed graph not fit the requirements for the subsequent decomposition, leading to compromised clustering accuracy. To this end, we propose a joint learning framework, which is able to learn the graph and the clustering result simultaneously, such that the resulting graph is tailored to the clustering task. The proposed model is formulated as a well-defined nonnegative and off-diagonal constrained optimization problem, which is further efficiently solved with convergence theoretically guaranteed. The advantage of the proposed model is demonstrated by comparing with 19 state-of-the-art clustering methods on 10 datasets with 4 clustering metrics.

Collaborative representation is a popular feature learning approach, which encoding process is assisted by variety types of information. In this paper, we propose a collaborative representation restricted Boltzmann Machine (CRRBM) for modeling binary data and a collaborative representation Gaussian restricted Boltzmann Machine (CRGRBM) for modeling realvalued data by applying a collaborative representation strategy in the encoding procedure. We utilize Locality Sensitive Hashing (LSH) to generate similar sample subsets of the instance and observed feature set simultaneously from input data. Hence, we can obtain some mini blocks, which come from the intersection of instance and observed feature subsets. Then we integrate Contrastive Divergence and Bregman Divergence methods with mini blocks to optimize our CRRBM and CRGRBM models. In their training process, the complex collaborative relationships between multiple instances and features are fused into the hidden layer encoding. Hence, these encodings have dual characteristics of concealment and cooperation. Here, we develop two deep collaborative encoder frameworks (DCEF) based on the CRRBM and CRGRBM models: one is a DCEF with Gaussian linear visible units (GDCEF) for modeling real-valued data, and the other is a DCEF with binary visible units (BDCEF) for modeling binary data. We explore the collaborative representation capability of the hidden features in every layer of the GDCEF and BDCEF framework, especially in the deepest hidden layer. The experimental results show that the GDCEF and BDCEF frameworks have more outstanding performances than the classic Autoencoder framework for unsupervised clustering task on the MSRA-MM2.0 and UCI datasets, respectively.

Local graph clustering methods aim to find small clusters in very large graphs. These methods take as input a graph and a seed node, and they return as output a good cluster in a running time that depends on the size of the output cluster but that is independent of the size of the input graph. In this paper, we adopt a statistical perspective on local graph clustering, and we analyze the performance of the l1-regularized PageRank method (a popular local graph clustering method) for the recovery of a single target cluster, given a seed node inside the cluster. Assuming the target cluster has been generated by a random model, we present two results. In the first, we show that the optimal support of l1-regularized PageRank recovers the full target cluster, with bounded false positives. In the second, we show that if the seed node is connected solely to the target cluster then the optimal support of l1-regularized PageRank recovers exactly the target cluster. We also show that the solution path of l1-regularized PageRank is monotonic. From a computational perspective, this permits the application of the forward stagewise algorithm, which in turn permits us to approximate the entire solution path of the local cluster in a running time that does not depend on the size of the entire graph.

An $\varepsilon$-coreset for a given set $D$ of $n$ points, is usually a small weighted set, such that querying the coreset \emph{provably} yields a $(1+\varepsilon)$-factor approximation to the original (full) dataset, for a given family of queries. Using existing techniques, coresets can be maintained for streaming, dynamic (insertion/deletions), and distributed data in parallel, e.g. on a network, GPU or cloud. We suggest the first coresets that approximate the negative log-likelihood for $k$-Gaussians Mixture Models (GMM) of arbitrary shapes (ratio between eigenvalues of their covariance matrices). For example, for any input set $D$ whose coordinates are integers in $[-n^{100},n^{100}]$ and any fixed $k,d\geq 1$, the coreset size is $(\log n)^{O(1)}/\varepsilon^2$, and can be computed in time near-linear in $n$, with high probability. The optimal GMM may then be approximated quickly by learning the small coreset. Previous results [NIPS'11, JMLR'18] suggested such small coresets for the case of semi-speherical unit Gaussians, i.e., where their corresponding eigenvalues are constants between $\frac{1}{2\pi}$ to $2\pi$. Our main technique is a reduction between coresets for $k$-GMMs and projective clustering problems. We implemented our algorithms, and provide open code, and experimental results. Since our coresets are generic, with no special dependency on GMMs, we hope that they will be useful for many other functions.

We consider the problem of identifying multiway block structure from a large noisy tensor. Such problems arise frequently in applications such as genomics, recommendation system, topic modeling, and sensor network localization. We propose a tensor block model, develop a unified least-square estimation, and obtain the theoretical accuracy guarantees for multiway clustering. The statistical convergence of the estimator is established, and we show that the associated clustering procedure achieves partition consistency. A sparse regularization is further developed for identifying important blocks with elevated means. The proposal handles a broad range of data types, including binary, continuous, and hybrid observations. Through simulation and application to two real datasets, we demonstrate the outperformance of our approach over previous methods.

Minimax linkage was first introduced by Ao et al. [3] in 2004, as an alternative to standard linkage methods used in hierarchical clustering. Minimax linkage relies on distances to a prototype for each cluster; this prototype can be thought of as a representative object in the cluster, hence improving the interpretability of clustering results. Bien and Tibshirani analyzed properties of this method in 2011 [2], popularizing the method within the statistics community. Additionally, they performed comparisons of minimax linkage to standard linkage methods, making use of five data sets and two different evaluation metrics (distance to prototype and misclassification rate). In an effort to expand upon their work and evaluate minimax linkage more comprehensively, our benchmark study focuses on thorough method evaluation via multiple performance metrics on several well-described data sets. We also make all code and data publicly available through an R package, for full reproducibility. Similarly to [2], we find that minimax linkage often produces the smallest maximum minimax radius of all linkage methods, meaning that minimax linkage produces clusters where objects in a cluster are tightly clustered around their prototype. This is true across a range of values for the total number of clusters (k). However, this is not always the case, and special attention should be paid to the case when k is the true known value. For true k, minimax linkage does not always perform the best in terms of all the evaluation metrics studied, including maximum minimax radius. This paper was motivated by the IFCS Cluster Benchmarking Task Force's call for clustering benchmark studies and the white paper [5], which put forth guidelines and principles for comprehensive benchmarking in clustering. Our work is designed to be a neutral benchmark study of minimax linkage.

For the degree corrected stochastic block model in the presence of arbitrary or even adversarial outliers, we develop a convex-optimization-based clustering algorithm that includes a penalization term depending on the positive deviation of a node from the expected number of edges to other inliers. We prove that under mild conditions, this method achieves exact recovery of the underlying clusters. Our synthetic experiments show that our algorithm performs well on heterogeneous networks, and in particular those with Pareto degree distributions, for which outliers have a broad range of possible degrees that may enhance their adversarial power. We also demonstrate that our method allows for recovery with significantly lower error rates compared to existing algorithms.