New Paradigms and Optimality Guarantees in Statistical Learning and Estimation

Machine learning (ML) has become one of the most powerful classes of tools for artificial intelligence, personalized web services and data science problems across fields. Within the field of machine learning itself, there had been quite a number of paradigm shifts caused by the explosion of data size, computing power, modeling tools, and the new ways people collect, share, and make use of data sets. Data privacy, for instance, was much less of a problem before the availability of personal information online that could be used to identify users in anonymized data sets. Images, videos, as well as observations generated over a social networks, often have highly localized structures, that cannot be captured by standard nonparametric models. Moreover, the “common task framework” that is adopted by many sub- disciplines of AI has made it possible for many people to collaboratively and repeated work on the same data set, leading to implicit overfitting on public benchmarks. In addition, data collected in many internet services, e.g., web search and targeted ads, are not iid, but rather feedbacks specific to the deployed algorithm. This thesis presents technical contributions under a number of new mathematical frameworks that are designed to partially address these new paradigms. • Firstly, we consider the problem of statistical learning with privacy constraints. Under Vapnik’s general learning setting and the formalism of differential privacy (DP), we establish simple conditions that characterizes the private learnability, which reveals a mixture of positive and negative insight. We then identify generic methods that reuses existing randomness to effectively solve private learning in practice; and discuss weaker notions of privacy that allows for more favorable privacy-utility tradeoff. • Secondly, we develop a few generalizations of trend filtering, a locally-adaptive nonparametric regression technique that is minimax in 1D, to the multivariate setting and to graphs. We also study specific instances of the problems, e.g., total variation denoising on d-dimensional grids more closely and the results reveal interesting statistical computational trade-offs. • Thirdly, we investigate two problems in sequential interactive learning: a) off- policy evaluation in contextual bandits, that aims to use data collected from one algorithm to evaluate the performance of a different algorithm; b) the problem of adaptive data analysis, that uses randomization to prevent adversarial data analysts from a form of “p-hacking” through multiple steps of sequential data access. In the above problems, we will provide not only performance guarantees of algorithms but also certain notions of optimality. Whenever applicable, careful empirical studies on synthetic and real data are also included.