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Explore advanced mathematical techniques for analyzing iterative optimization algorithms in high-dimensional settings through this probability and statistics seminar lecture. Discover a comprehensive toolbox for understanding the dynamics of both convex and non-convex optimization methods when working with random data, covering first-order methods like gradient descent and approximate message passing, as well as higher-order approaches including proximal point methods, prox-linear techniques, alternating minimization, and expectation maximization variants. Learn how to obtain exact deterministic descriptions of algorithmic behavior and establish finite-sample guarantees that bound deviations between empirical iterates and their theoretical counterparts. Examine the mathematical foundations built on sequential variants of Gordon's Gaussian comparison inequalities combined with Bolthausen's Gaussian conditioning technique, providing rigorous theoretical frameworks for algorithm analysis in probabilistic settings.
