Robust estimation via generalized quasi-gradients

• URL: http://arxiv.org/abs/2005.14073v1
• Date: Thu, 28 May 2020 15:14:33 GMT
• Title: Robust estimation via generalized quasi-gradients
• Authors: Banghua Zhu, Jiantao Jiao and Jacob Steinhardt
• Abstract summary: We show why many recently proposed robust estimation problems are efficiently solvable. We identify the existence of "generalized quasi-gradients" We show that generalized quasi-gradients exist and construct efficient algorithms.
• Score: 28.292300073453877
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\epsilon < 1/3$. 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 $O(\sqrt{\epsilon})$ to the optimal rate of $O(\epsilon)$ for small $\epsilon$ 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 $\tilde{O}(d/\epsilon^2)$.

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