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Statistics meets Optimization: Computational guarantees for statistical learning algorithms


Modern technological advances have prompted massive scale data collection in many

modern fields such as artificial intelligence, and traditional sciences alike. This has led to

an increasing need for scalable machine learning algorithms and statistical methods to draw

conclusions about the world. In all data-driven procedures, the data scientist faces the

following fundamental questions: How should I design the learning algorithm and how long

should I run it? Which samples should I collect for training and how many are sufficient to

generalize conclusions to unseen data? These questions relate to statistical and computational properties of both the data and the algorithm. This thesis explores their role in the areas of non-convex optimization, non-parametric estimation, active learning and multiple testing.

In the first part, we provide insights of different flavor concerning the

interplay between statistical and computational properties of first-order type methods on

common estimation procedures. The expectation-maximization (EM) algorithm estimates

parameters of a latent variable model by running a first-order type method on a non-convex

landscape. We identify and characterize a general class of Hidden Markov Models for which

linear convergence of EM to a statistically optimal point is provable for a large initialization

radius. For non-parametric estimation problems, functional gradient descent type (also

called boosting) algorithms are used to estimate the best fit in infinite dimensional function

spaces. We develop a new proof technique showing that early stopping the algorithm instead

may also yield an optimal estimator without explicit regularization. In fact, the same

key quantities (localized complexities) are underlying both traditional penalty-based and

algorithmic regularization.

In the second part of the thesis, we explore how data collected adaptively with a constantly

updated estimation can lead to signifcant reduction in sample complexity for multiple

hypothesis testing problems. In particular, we show how adaptive strategies can be used

to simultaneously control the false discovery rate over multiple tests and return the best

alternative (among many) for each test with optimal sample complexity in an online manner.

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