High-Dimensional Robust Mean Estimation via Gradient Descent
This work addresses robust statistics for data analysis in noisy environments, establishing a connection to non-convex optimization, but it appears incremental as it builds on existing polynomial-time algorithms.
The paper tackles the problem of high-dimensional robust mean estimation with adversarial outliers by showing that gradient descent can solve a non-convex formulation, achieving near-optimal solutions through a novel structural lemma.
We study the problem of high-dimensional robust mean estimation in the presence of a constant fraction of adversarial outliers. A recent line of work has provided sophisticated polynomial-time algorithms for this problem with dimension-independent error guarantees for a range of natural distribution families. In this work, we show that a natural non-convex formulation of the problem can be solved directly by gradient descent. Our approach leverages a novel structural lemma, roughly showing that any approximate stationary point of our non-convex objective gives a near-optimal solution to the underlying robust estimation task. Our work establishes an intriguing connection between algorithmic high-dimensional robust statistics and non-convex optimization, which may have broader applications to other robust estimation tasks.