Year: 2024
Type: article
Abstract: We give Hoeffding- and Bernstein-type concentration inequalities for the largest eigenvalue of sums of random matrices arising from a Markov chain. We consider time-dependent matrix-valued functions on a general state space, generalizing previous results that had only considered Hoeffding-type inequalities, and only for time-independent functions on a finite state space. In particular, we study a kind of non-commutative moment generating function, provide tight bounds on this object and use a method of Garg et al. (A matrix expander Chernoff bound. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, page 1102–1114, New York, NY, USA, 2018. Association for Computing Machinery) to turn this into tail bounds. Our proof proceeds spectrally, bounding the norm of a certain perturbed operator. In the process we make an interesting connection to dynamical systems and Banach space theory to prove a crucial result on the limiting behaviour of our moment generating function that may be of independent interest.
Authors: Joe Neeman, Bobby Shi, Rachel Ward
Institution: The University of Texas at Austin
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Isoperimetric Clusters and Related Extremal Problems with Applications in Probability, EXC 2047: Hausdorff Center for Mathematics (HCM), G1523888516, G1600773818, CAREER: Learning, testing, and hardness via extremal geometric problems, G3474365249, FRG: Collaborative Research: Randomized Algorithms for Solving Linear Systems, G591746060, G6562578720, G6671297155, Learning with Confidence: Bootstrapping Error Estimates for Stochastic Iterative Algorithms, AI Institute: Institute for Foundations of Machine Learning
OpenAlex ID: W4405228575