<p dir="ltr">Continuous optimization is a fundamental topic both in theoretical computer science and applications of machine learning. Meanwhile, an important idea in the development modern algorithms it the use of randomness to achieve empirical speedup and improved theoretical runtimes. Stochastic gradient descent (SGD) and matrix-multiplication time linear program solvers [1] are two important examples of such achievements. Matrix sketching and related ideas provide a theoretical framework for the behavior of random matrices and vectors that arise in these algorithms, thereby provide a natural way to better understand the behavior of such randomized algorithms. In this dissertation, we consider three general problems in this area.</p>

<p>This work explores how randomization can be exploited to deliver sophisticated</p><p>algorithms with provable bounds for: (i) The approximation of matrix functions, such</p><p>as the log-determinant and the Von-Neumann entropy; and (ii) The low-rank approximation</p><p>of matrices. Our algorithms are inspired by recent advances in Randomized</p><p>Numerical Linear Algebra (RandNLA), an interdisciplinary research area that exploits</p><p>randomization as a computational resource to develop improved algorithms for</p><p>large-scale linear algebra problems. The main goal of this work is to encourage the</p><p>practical use of RandNLA approaches to solve Big Data bottlenecks at industrial</p><p>level. Our extensive evaluation tests are complemented by a thorough theoretical</p><p>analysis that proves the accuracy of the proposed algorithms and highlights their</p><p>scalability as the volume of data increases. Finally, the low computational time and</p><p>memory consumption, combined with simple implementation schemes that can easily</p><p>be extended in parallel and distributed environments, render our algorithms suitable</p><p>for use in the development of highly efficient real-world software.</p>