- Tu Trinh
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Abstract:
Many statistical estimators are defined as the fixed point of a data-dependent operator, with estimators based on minimizing a cost function being an important special case. The limiting performance of such estimators depends on the properties of the population-level operator in the idealized limit of infinitely many samples. We develop a general framework that yields bounds on statistical accuracy based on the interplay between the deterministic convergence rate of the algorithm at the population level, and its degree of (in)stability when applied to an empirical object based on n samples. Using this framework, we analyze both stable forms of gradient descent and some higher-order and unstable algorithms, including Newton’s method and its cubic-regularized variant, as well as the EM algorithm. We provide applications of our general results to several concrete classes of singular statistical models, including Gaussian mixture estimation, single-index models, and informative non-response models. We exhibit cases in which an unstable algorithm can achieve the same statistical accuracy as a stable algorithm in exponentially fewer steps—namely, with the number of iterations being reduced from polynomial to logarithmic in sample size n.
About our speaker:
Nhat Ho is currently an Assistant Professor of Data Science, Machine Learning, and Statistics at the University of Texas at Austin. He is a core member of the University of Texas Austin Machine Learning Laboratory and senior personnel of the Institute for Foundations of Machine Learning. His current research focuses on the interplay of four principles of machine learning, statistics, and data science: heterogeneity of data (Bayesian nonparametrics, mixture and hierarchical models), interpretability of complex and deep models (deep generative models, convolutional neural networks, Transformer, model misspecification), stability, and scalability of optimization and sampling algorithms (computational optimal transport, (non)-convex optimization in statistical settings, sampling and variational inference).
