Beyond the Worst-Case Analysis of Algorithms

Learning in the presence of outliers is a fundamental problem in statistics. Until recently, all known efficient unsupervised learning algorithms were very sensitive to outliers in high dimensions. In particular, even for the task of robust mean estimation under natural distributional assumptions, no efficient algorithm was known. A recent line of work gave the first efficient robust estimators for a number of fundamental statistical tasks, including mean and covariance estimation. This chapter introduces the core ideas and techniques in the emerging area of algorithmic highdimensional robust statistics with a focus on robust mean estimation.