A Scalable and Extensible Framework for Superposition-Structured Models

In many learning tasks, structural models usually lead to better interpretability and higher generalization performance. In recent years, however, the simple structural models such as lasso are frequently proved to be insufficient. Accordingly, there has been a lot of work on "superposition-structured" models where multiple structural constraints are imposed. To efficiently solve these "superposition-structured" statistical models, we develop a framework based on a proximal Newton-type method. Employing the smoothed conic dual approach with the LBFGS updating formula, we propose a scalable and extensible proximal quasi-Newton (SEP-QN) framework. Empirical analysis on various datasets shows that our framework is potentially powerful, and achieves super-linear convergence rate for optimizing some popular "superposition-structured" statistical models such as the fused sparse group lasso

Wishart Mechanism for Differentially Private Principal Components Analysis

We propose a new input perturbation mechanism for publishing a covariance matrix to achieve $(\epsilon,0)$-differential privacy. Our mechanism uses a Wishart distribution to generate matrix noise. In particular, We apply this mechanism to principal component analysis. Our mechanism is able to keep the positive semi-definiteness of the published covariance matrix. Thus, our approach gives rise to a general publishing framework for input perturbation of a symmetric positive semidefinite matrix. Moreover, compared with the classic Laplace mechanism, our method has better utility guarantee. To the best of our knowledge, Wishart mechanism is the best input perturbation approach for $(\epsilon,0)$-differentially private PCA. We also compare our work with previous exponential mechanism algorithms in the literature and provide near optimal bound while having more flexibility and less computational intractability.Comment: A full version with technical proofs. Accepted to AAAI-1