Robust estimation via generalized quasi-gradients

We explore why many recently proposed robust estimation problems are efficiently solvable, even though the underlying optimization problems are non-convex. We study the loss landscape of these robust estimation problems, and identify the existence of "generalized quasi-gradients". Whenever these quasi-gradients exist, a large family of low-regret algorithms are guaranteed to approximate the global minimum; this includes the commonly-used filtering algorithm. For robust mean estimation of distributions under bounded covariance, we show that any first-order stationary point of the associated optimization problem is an {approximate global minimum} if and only if the corruption level $ε< 1/3$. Consequently, any optimization algorithm that aproaches a stationary point yields an efficient robust estimator with breakdown point $1/3$. With careful initialization and step size, we improve this to $1/2$, which is optimal. For other tasks, including linear regression and joint mean and covariance estimation, the loss landscape is more rugged: there are stationary points arbitrarily far from the global minimum. Nevertheless, we show that generalized quasi-gradients exist and construct efficient algorithms. These algorithms are simpler than previous ones in the literature, and for linear regression we improve the estimation error from $O(\sqrtε)$ to the optimal rate of $O(ε)$ for small $ε$ assuming certified hypercontractivity. For mean estimation with near-identity covariance, we show that a simple gradient descent algorithm achieves breakdown point $1/3$ and iteration complexity $\tilde{O}(d/ε^2)$.

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