  We study high-dimensional robust statistics tasks in the streaming model. A
recent line of work obtained computationally efficient algorithms for a range
of high-dimensional robust estimation tasks. Unfortunately, all previous
algorithms require storing the entire dataset, incurring memory at least
quadratic in the dimension. In this work, we develop the first efficient
streaming algorithms for high-dimensional robust statistics with near-optimal
memory requirements (up to logarithmic factors). Our main result is for the
task of high-dimensional robust mean estimation in (a strengthening of) Huber's
contamination model. We give an efficient single-pass streaming algorithm for
this task with near-optimal error guarantees and space complexity nearly-linear
in the dimension. As a corollary, we obtain streaming algorithms with
near-optimal space complexity for several more complex tasks, including robust
covariance estimation, robust regression, and more generally robust stochastic
optimization.
