Cubic-Regularized Newton for Spectral Constrained Matrix Optimization and its Application to Fairness

• URL: http://arxiv.org/abs/2209.01229v1
• Date: Fri, 2 Sep 2022 18:11:05 GMT
• Title: Cubic-Regularized Newton for Spectral Constrained Matrix Optimization and its Application to Fairness
• Authors: Casey Garner, Gilad Lerman, Shuzhong Zhang
• Abstract summary: Matrix functions are utilized to rewrite smooth spectral constrained matrix optimization problems. A new convergence analysis is provided for cubic-regularized Newton for matrix vector spaces.
• Score: 9.649070872824957
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Matrix functions are utilized to rewrite smooth spectral constrained matrix optimization problems as smooth unconstrained problems over the set of symmetric matrices which are then solved via the cubic-regularized Newton method. A second-order chain rule identity for matrix functions is proven to compute the higher-order derivatives to implement cubic-regularized Newton, and a new convergence analysis is provided for cubic-regularized Newton for matrix vector spaces. We demonstrate the applicability of our approach by conducting numerical experiments on both synthetic and real datasets. In our experiments, we formulate a new model for estimating fair and robust covariance matrices in the spirit of the Tyler's M-estimator (TME) model and demonstrate its advantage.

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