Network Lasso: Clustering and Optimization in Large Graphs (1507.00280v1)

Abstract: Convex optimization is an essential tool for modern data analysis, as it provides a framework to formulate and solve many problems in machine learning and data mining. However, general convex optimization solvers do not scale well, and scalable solvers are often specialized to only work on a narrow class of problems. Therefore, there is a need for simple, scalable algorithms that can solve many common optimization problems. In this paper, we introduce the \emph{network lasso}, a generalization of the group lasso to a network setting that allows for simultaneous clustering and optimization on graphs. We develop an algorithm based on the Alternating Direction Method of Multipliers (ADMM) to solve this problem in a distributed and scalable manner, which allows for guaranteed global convergence even on large graphs. We also examine a non-convex extension of this approach. We then demonstrate that many types of problems can be expressed in our framework. We focus on three in particular - binary classification, predicting housing prices, and event detection in time series data - comparing the network lasso to baseline approaches and showing that it is both a fast and accurate method of solving large optimization problems.

• The paper presents the network lasso formulation, extending group lasso to graph settings for simultaneous clustering and convex optimization.
• It develops a distributed ADMM solver that guarantees global convergence and scales effectively to large, complex datasets.
• Empirical results show improved classification accuracy, enhanced housing price predictions, and robust anomaly detection on time series data.

An Overview of Network Lasso: Clustering and Optimization in Large Graphs

The paper "Network Lasso: Clustering and Optimization in Large Graphs" by David Hallac, Jure Leskovec, and Stephen Boyd introduces the concept of the network lasso as a framework for simultaneous clustering and optimization on large graphs. This work generalizes the group lasso to network settings and establishes a unifying modeling and algorithmic approach to address a range of problems prevalent in machine learning and data mining.

Core Contributions

The paper's primary contributions can be summarized as follows:

• Formal Definition of Network Lasso: The authors formally define the network lasso problem as a specific type of convex optimization problem suited for graphs. This formulation extends the group lasso, traditionally used in feature selection, to handle networks, thereby facilitating clustering alongside optimization tasks.
• Development of a Distributed and Scalable Solver: Utilizing the Alternating Direction Method of Multipliers (ADMM), the paper presents an efficient and scalable algorithm for solving the network lasso problem. This solver is designed for large datasets, operating in a distributed manner with guaranteed global convergence.
• Practical Applications Highlighted: The network lasso framework applies to numerous common scenarios, such as binary classification, housing price prediction, and event detection in time series data, demonstrating its versatility in real-world applications.

Numerical Results and Claims

The numerical experiments conducted provide compelling evidence of the network lasso's capability to outperform baseline methods in various tasks:

• Improved Classification Accuracy: In a synthetic dataset designed for binary classification, the network lasso achieved significant improvements in prediction accuracy compared to both local and global SVM approaches, highlighting its strength in extracting and leveraging network structure for improved model performance.
• Enhanced Housing Price Predictions: When predicting housing prices, the network lasso effectively captured neighborhood structures, yielding lower mean squared errors compared to traditional linear regression models. This example underscores the method's ability to handle geographic clustering where conventional models fail.
• Effective Anomaly Detection: The network lasso demonstrated robust performance in detecting events within time series data by modeling traffic anomalies in a building access dataset, outperforming a Poisson-based baseline in correctly identifying events.

Implications and Future Directions

The network lasso's ability to transform large-scale optimization problems into tractable distributed algorithms opens up significant theoretical and practical implications. From a theoretical standpoint, this work suggests promising directions for further refinement of clustering and regularization techniques within the ADMM framework, particularly in developing more sophisticated strategies for selecting the regularization parameter $\lambda$.

Practically, the framework can be immediately applied to diverse problem domains, such as sensor networks, recommendation systems, and financial markets, where network structures naturally arise. The possibility of integrating non-convex extensions, which approximate $\ell_0$ norms, also presents interesting avenues for research in achieving sparser solutions.

In conclusion, the network lasso offers a compelling and flexible approach to tackling a broad class of optimization problems on graphs, grounded in a rigorous mathematical framework and supported by extensive empirical evaluation. Future work may extend this framework's applicability and optimize its computational performance to adapt to even larger, more complex datasets and networks.