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We study constrained nonconvex optimization problems in machine learning and signal processing. It is well-known that these problems can be rewritten to a min-max problem in a Lagrangian form. However, due to the lack of convexity, their landscape is not well understood and how to find the stable equilibria of the Lagrangian function is still unknown. To bridge the gap, we study the landscape of the Lagrangian function. Further, we define a special class of Lagrangian functions. They enjoy the following two properties: 1. Equilibria are either stable or unstable (Formal definition in Section 2); 2.Stable equilibria correspond to the global optima of the original problem. We show that a generalized eigenvalue (GEV) problem, including canonical correlation analysis and other problems as special examples, belongs to the class. Specifically, we characterize its stable and unstable equilibria by leveraging an invariant group and symmetric property (more details in Section 3). Motivated by these neat geometric structures, we propose a simple, efficient, and stochastic primal-dual algorithm solving the online GEV problem. Theoretically, under sufficient conditions, we establish an asymptotic rate of convergence and obtain the first sample complexity result for the online GEV problem by diffusion approximations, which are widely used in applied probability. Numerical results are also provided to support our theory. 

Many machine learning techniques sacrifice convenient computational structures to gain estimation robustness and modeling flexibility. However, by exploring the modeling structures, we find these ``sacrifices'' do not always require more computational efforts. To shed light on such a ``free-lunch'' phenomenon, we study the square-root-Lasso (SQRT-Lasso) type regression problem. Specifically, we show that the nonsmooth loss functions of SQRT-Lasso type regression ease tuning effort and gain adaptivity to inhomogeneous noise, but is not necessarily more challenging than Lasso in computation. We can directly apply proximal algorithms (e.g. proximal gradient descent, proximal Newton, and proximal quasi-Newton algorithms) without worrying about the nonsmoothness of the loss function. Theoretically, we prove that the proximal algorithms combined with the pathwise optimization scheme enjoy fast convergence guarantees with high probability. Numerical results are provided to support our theory. 

Abstract The semiconductor tracker (SCT) is one of the tracking systems for charged particles in the ATLAS detector. It consists of 4088 silicon strip sensor modules.During Run 2 (2015–2018) the Large Hadron Collider delivered an integrated luminosity of 156 fb -1 to the ATLAS experiment at a centre-of-mass proton-proton collision energy of 13 TeV. The instantaneous luminosity and pile-up conditions were far in excess of those assumed in the original design of the SCT detector.Due to improvements to the data acquisition system, the SCT operated stably throughout Run 2.It was available for 99.9% of the integrated luminosity and achieved a data-quality efficiency of 99.85%.Detailed studies have been made of the leakage current in SCT modules and the evolution of the full depletion voltage, which are used to study the impact of radiation damage to the modules.