Splitting Method for Support Vector Machine in Reproducing Kernel Banach Space with Lower Semi-continuous Loss Function

In this paper, we use the splitting method to solve support vector machine in reproducing kernel Banach space with lower semi-continuous loss function. We equivalently transfer support vector machines in reproducing kernel Banach space with lower semi-continuous loss function to a finite-dimensional tensor Optimization and propose the splitting method based on alternating direction method of multipliers. By Kurdyka-Lojasiewicz inequality, the iterative sequence obtained by this splitting method is globally convergent to a stationary point if the loss function is lower semi-continuous and subanalytic. Finally, several numerical performances demonstrate the effectiveness.Comment: arXiv admin note: text overlap with arXiv:2208.1252

Quantized Consensus ADMM for Multi-Agent Distributed Optimization

Multi-agent distributed optimization over a network minimizes a global objective formed by a sum of local convex functions using only local computation and communication. We develop and analyze a quantized distributed algorithm based on the alternating direction method of multipliers (ADMM) when inter-agent communications are subject to finite capacity and other practical constraints. While existing quantized ADMM approaches only work for quadratic local objectives, the proposed algorithm can deal with more general objective functions (possibly non-smooth) including the LASSO. Under certain convexity assumptions, our algorithm converges to a consensus within $\log_{1+\eta}\Omega$ iterations, where $\eta>0$ depends on the local objectives and the network topology, and $\Omega$ is a polynomial determined by the quantization resolution, the distance between initial and optimal variable values, the local objective functions and the network topology. A tight upper bound on the consensus error is also obtained which does not depend on the size of the network.Comment: 30 pages, 4 figures; to be submitted to IEEE Trans. Signal Processing. arXiv admin note: text overlap with arXiv:1307.5561 by other author

Recovery Conditions and Sampling Strategies for Network Lasso

The network Lasso is a recently proposed convex optimization method for machine learning from massive network structured datasets, i.e., big data over networks. It is a variant of the well-known least absolute shrinkage and selection operator (Lasso), which is underlying many methods in learning and signal processing involving sparse models. Highly scalable implementations of the network Lasso can be obtained by state-of-the art proximal methods, e.g., the alternating direction method of multipliers (ADMM). By generalizing the concept of the compatibility condition put forward by van de Geer and Buehlmann as a powerful tool for the analysis of plain Lasso, we derive a sufficient condition, i.e., the network compatibility condition, on the underlying network topology such that network Lasso accurately learns a clustered underlying graph signal. This network compatibility condition relates the location of the sampled nodes with the clustering structure of the network. In particular, the NCC informs the choice of which nodes to sample, or in machine learning terms, which data points provide most information if labeled.Comment: nominated as student paper award finalist at Asilomar 2017. arXiv admin note: substantial text overlap with arXiv:1704.0210

Proximal Symmetric Non-negative Latent Factor Analysis: A Novel Approach to Highly-Accurate Representation of Undirected Weighted Networks

An Undirected Weighted Network (UWN) is commonly found in big data-related applications. Note that such a network's information connected with its nodes, and edges can be expressed as a Symmetric, High-Dimensional and Incomplete (SHDI) matrix. However, existing models fail in either modeling its intrinsic symmetry or low-data density, resulting in low model scalability or representation learning ability. For addressing this issue, a Proximal Symmetric Nonnegative Latent-factor-analysis (PSNL) model is proposed. It incorporates a proximal term into symmetry-aware and data density-oriented objective function for high representation accuracy. Then an adaptive Alternating Direction Method of Multipliers (ADMM)-based learning scheme is implemented through a Tree-structured of Parzen Estimators (TPE) method for high computational efficiency. Empirical studies on four UWNs demonstrate that PSNL achieves higher accuracy gain than state-of-the-art models, as well as highly competitive computational efficiency

Iteration Complexity Analysis of Multi-Block ADMM for a Family of Convex Minimization without Strong Convexity

The alternating direction method of multipliers (ADMM) is widely used in solving structured convex optimization problems due to its superior practical performance. On the theoretical side however, a counterexample was shown in [7] indicating that the multi-block ADMM for minimizing the sum of $N$ $(N\geq 3)$ convex functions with $N$ block variables linked by linear constraints may diverge. It is therefore of great interest to investigate further sufficient conditions on the input side which can guarantee convergence for the multi-block ADMM. The existing results typically require the strong convexity on parts of the objective. In this paper, we present convergence and convergence rate results for the multi-block ADMM applied to solve certain $N$-block $(N\geq 3)$ convex minimization problems without requiring strong convexity. Specifically, we prove the following two results: (1) the multi-block ADMM returns an $\epsilon$-optimal solution within $O(1/\epsilon^2)$ iterations by solving an associated perturbation to the original problem; (2) the multi-block ADMM returns an $\epsilon$-optimal solution within $O(1/\epsilon)$ iterations when it is applied to solve a certain sharing problem, under the condition that the augmented Lagrangian function satisfies the Kurdyka-Lojasiewicz property, which essentially covers most convex optimization models except for some pathological cases.Comment: arXiv admin note: text overlap with arXiv:1408.426