In the first part of this dissertation, we consider two problems in sequential decision making. The first problem we consider is sequential selection of a monotone subsequence from a random permutation. We find a two term asymptotic expansion for the optimal expected value of a sequentially selected monotone subsequence from a random permutation of length n. The second problem we consider deals with the multiplicative relaxation or constriction of the classical problem of the number of records in a sequence of n independent and identically distributed observations. In the relaxed case, we find a central limit theorem (CLT) with a different normalization than Renyi's classical CLT, and in the constricted case we find convergence in distribution to an unbounded random variable.;In the second part of this dissertation, we put forward two large-scale randomized algorithms. We propose a two-step sensing scheme for the low-rank matrix recovery problem which requires far less storage space and has much lower computational complexity than other state-of-art methods based on nuclear norm minimization. We introduce a fast iterative reweighted least squares algorithm, textit{Guluru}, based on subsampled randomized Hadamard transform, to solve a wide class of generalized linear models.
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