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Register a new userA Complete Analysis of the l_1,p Group-Lasso
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Julia Vogt
Publication date
2012
fields
Informatics Engineering
and research's language is
English
Authors
Julia Vogt
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The Group-Lasso is a well-known tool for joint regularization in machine learning methods. While the l_{1,2} and the l_{1,infty} version have been studied in detail and efficient algorithms exist, there are still open questions regarding other l_{1,p} variants. We characterize conditions for solutions of the l_{1,p} Group-Lasso for all p-norms with 1 <= p <= infty, and we present a unified active set algorithm. For all p-norms, a highly efficient projected gradient algorithm is presented. This new algorithm enables us to compare the prediction performance of many variants of the Group-Lasso in a multi-task learning setting, where the aim is to solve many learning problems in parallel which are coupled via the Group-Lasso constraint. We conduct large-scale experiments on synthetic data and on two real-world data sets. In accordance with theoretical characterizations of the different norms we observe that the weak-coupling norms with p between 1.5 and 2 consistently outperform the strong-coupling norms with p >> 2.
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Many applications generate data with an intrinsic network structure such as time series data, image data or social network data. The network Lasso (nLasso) has been proposed recently as a method for joint clustering and optimization of machine learning models for networked data. The nLasso extends the Lasso from sparse linear models to clustered graph signals. This paper explores the duality of nLasso and network flow optimization. We show that, in a very precise sense, nLasso is equivalent to a minimum-cost flow problem on the data network structure. Our main technical result is a concise characterization of nLasso solutions via existence of certain network flows. The main conceptual result is a useful link between nLasso methods and basic graph algorithms such as clustering or maximum flow.
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Although the optimization objectives for learning neural networks are highly non-convex, gradient-based methods have been wildly successful at learning neural networks in practice. This juxtaposition has led to a number of recent studies on provable guarantees for neural networks trained by gradient descent. Unfortunately, the techniques in these works are often highly specific to the problem studied in each setting, relying on different assumptions on the distribution, optimization parameters, and network architectures, making it difficult to generalize across different settings. In this work, we propose a unified non-convex optimization framework for the analysis of neural network training. We introduce the notions of proxy convexity and proxy Polyak-Lojasiewicz (PL) inequalities, which are satisfied if the original objective function induces a proxy objective function that is implicitly minimized when using gradient methods. We show that stochastic gradient descent (SGD) on objectives satisfying proxy convexity or the proxy PL inequality leads to efficient guarantees for proxy objective functions. We further show that many existing guarantees for neural networks trained by gradient descent can be unified through proxy convexity and proxy PL inequalities.
A generalized gamification framework is introduced as a form of smart infrastructure with potential to improve sustainability and energy efficiency by leveraging humans-in-the-loop strategy. The proposed framework enables a Human-Centric Cyber-Physical System using an interface to allow building managers to interact with occupants. The interface is designed for occupant engagement-integration supporting learning of their preferences over resources in addition to understanding how preferences change as a function of external stimuli such as physical control, time or incentives. Towards intelligent and autonomous incentive design, a noble statistical learning algorithm performing occupants energy usage behavior segmentation is proposed. We apply the proposed algorithm, Graphical Lasso, on energy resource usage data by the occupants to obtain feature correlations--dependencies. Segmentation analysis results in characteristic clusters demonstrating different energy usage behaviors. The features--factors characterizing human decision-making are made explainable.
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