ABSTRACT We apply network Lasso to solve binary classification (clustering) problems on network structured data. To this end, we generalize ordinary logistic regression to non-Euclidean data defined over a complex network structure. A scalable classification algorithm is obtained by applying the alternating direction methods of multipliers to solve this optimization problem. Index Terms-- compressed sensing, big data over networks, semi-supervised learning, classification, clustering, complex networks, convex optimization I. INTRODUCTION We consider the problem of classifying or clustering a large set of data points which conform to an underlying network structure. Such network-structured datasets arise in a wide range of application domains, e.g., image-and video processing as well as social networks [1].

This paper presents a novel Differential Evolution algorithm for protein folding optimization that is applied to a three-dimensional AB off-lattice model. The proposed algorithm includes two new mechanisms. A local search is used to improve convergence speed and to reduce the runtime complexity of the energy calculation. For this purpose, a local movement is introduced within the local search. The designed evolutionary algorithm has fast convergence speed and, therefore, when it is trapped into the local optimum or a relatively good solution is located, it is hard to locate a better similar solution. The similar solution is different from the good solution in only a few components. A component reinitialization method is designed to mitigate this problem. Both the new mechanisms and the proposed algorithm were analyzed on well-known amino acid sequences that are used frequently in the literature. Experimental results show that the employed new mechanisms improve the efficiency of our algorithm and that the proposed algorithm is superior to other state-of-the-art algorithms. It obtained a hit ratio of 100% for sequences up to 18 monomers, within a budget of $10^{11}$ solution evaluations. New best-known solutions were obtained for most of the sequences. The existence of the symmetric best-known solutions is also demonstrated in the paper.

User and item features of side information are crucial for accurate recommendation. However, the large number of feature dimensions, e.g., usually larger than 10^7, results in expensive storage and computational cost. This prohibits fast recommendation especially on mobile applications where the computational resource is very limited. In this paper, we develop a generic feature-based recommendation model, called Discrete Factorization Machine (DFM), for fast and accurate recommendation. DFM binarizes the real-valued model parameters (e.g., float32) of every feature embedding into binary codes (e.g., boolean), and thus supports efficient storage and fast user-item score computation. To avoid the severe quantization loss of the binarization, we propose a convergent updating rule that resolves the challenging discrete optimization of DFM. Through extensive experiments on two real-world datasets, we show that 1) DFM consistently outperforms state-of-the-art binarized recommendation models, and 2) DFM shows very competitive performance compared to its real-valued version (FM), demonstrating the minimized quantization loss.

In this paper, we propose an active learning algorithm and models which can gradually learn individual's preference through pairwise comparisons. The active learning scheme aims at finding individual's most preferred choice with minimized number of pairwise comparisons. The pairwise comparisons are encoded into probabilistic models based on assumptions of choice models and deep Gaussian processes. The next-to-compare decision is determined by a novel acquisition function. We benchmark the proposed algorithm and models using functions with multiple local optima and one public airline itinerary dataset. The experiments indicate the effectiveness of our active learning algorithm and models.

Network structure optimization is a fundamental task in complex network analysis. However, almost all the research on Bayesian optimization is aimed at optimizing the objective functions with vectorial inputs. In this work, we first present a flexible framework, denoted graph Bayesian optimization, to handle arbitrary graphs in the Bayesian optimization community. By combining the proposed framework with graph kernels, it can take full advantage of implicit graph structural features to supplement explicit features guessed according to the experience, such as tags of nodes and any attributes of graphs. The proposed framework can identify which features are more important during the optimization process. We apply the framework to solve four problems including two evaluations and two applications to demonstrate its efficacy and potential applications.

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In recent years, many variance reduced algorithms for empirical risk minimization have been introduced. In contrast to vanilla SGD, these methods converge linearly on strong convex problems. To obtain the variance reduction, current methods either require frequent passes over the full data to recompute gradients---without making any progress during this time (like in SVRG), or they require memory of the same size as the input problem (like SAGA). In this work, we propose k-SVRG, an algorithm that interpolates between those two extremes: it makes best use of the available memory and in turn does avoid full passes over the data without making progress. We prove linear convergence of k-SVRG on strongly convex problems and convergence to stationary points on non-convex problems. Numerical experiments show the effectiveness of our method.

We study gradient-based optimization methods obtained by directly discretizing a second-order ordinary differential equation (ODE) related to the continuous limit of Nesterov's accelerated gradient method. When the function is smooth enough, we show that acceleration can be achieved by a stable discretization of the ODE using standard Runge-Kutta integrators. Specifically, we prove that under Lipschitz-gradient, convexity and order-$(s+2)$ differentiability assumptions, the sequence of iterates generated by discretizing the proposed second-order ODE converges to the optimal solution at a rate of $\mathcal{O}({N^{-2\frac{s}{s+1}}})$, where $s$ is the order of the Runge-Kutta numerical integrator. By increasing $s$, the convergence rate of our method approaches the optimal rate of $\mathcal{O}({N^{-2}})$. Furthermore, we introduce a new local flatness condition on the objective, according to which rates even faster than $\mathcal{O}(N^{-2})$ can be achieved with low-order integrators and only gradient information. Notably, this flatness condition is satisfied by several standard loss functions used in machine learning, and it may be of broader independent interest. We provide numerical experiments that verify the theoretical rates predicted by our results.

A majority of machine learning algorithms minimize empirical risk by solving a convex or non-convex optimization. Structured predictors solve combinatorial optimizations, and their learning algorithms solve hybrid optimizations. And new approaches for stochastic optimization have become integral in modern deep learning methodology. Students who take this course will study the latest knowledge and foundational concepts on optimization in machine learning, including theoretical analyses of optimization-based learning algorithms, theoretical bounds of discrete optimization for structured prediction, and recent discoveries about non-convex optimization methods. Class meets Tuesday and Thursday from 12:30 PM to 1:45 PM in the New Classroom Building (NCB) 110A.

Inspired by the matching of supply to demand in logistical problems, the optimal transport (or Monge--Kantorovich) problem involves the matching of probability distributions defined over a geometric domain such as a surface or manifold. In its most obvious discretization, optimal transport becomes a large-scale linear program, which typically is infeasible to solve efficiently on triangle meshes, graphs, point clouds, and other domains encountered in graphics and machine learning. Recent breakthroughs in numerical optimal transport, however, enable scalability to orders-of-magnitude larger problems, solvable in a fraction of a second. Here, we discuss advances in numerical optimal transport that leverage understanding of both discrete and smooth aspects of the problem. State-of-the-art techniques in discrete optimal transport combine insight from partial differential equations (PDE) with convex analysis to reformulate, discretize, and optimize transportation problems. The end result is a set of theoretically-justified models suitable for domains with thousands or millions of vertices. Since numerical optimal transport is a relatively new discipline, special emphasis is placed on identifying and explaining open problems in need of mathematical insight and additional research.