| In this dissertation, we present the results of our research on three topics, namely, the design and analysis of online convex optimization algorithms, convergence rate analysis of proximal gradient homotopy algorithm for structured convex problems, and application of computational methods for study of brain cells in the visual cortex of the primate brain. In our work on online optimization, we have developed a systematic approach with a clear connection to regret minimization for design and worst case analysis of online optimization algorithms. We apply this approach to online experiment design problems. Our results on the convergence rate analysis of proximal gradient homotopy algorithm extends the linear convergence rate of this algorithm from l1 norm to a general class of norms called decomposable norms. Our results on the clustering of cells in the visual cortical area V4 reveal new categories of neurons in this area. We discuss how online adaptive algorithms can be utilized for classification of these neurons in closed loop neurophysiological experiments. |
