Robust Distribution Learning with Local and Global Adversarial Corruptions

• URL: http://arxiv.org/abs/2406.06509v2
• Date: Mon, 24 Jun 2024 17:53:33 GMT
• Title: Robust Distribution Learning with Local and Global Adversarial Corruptions
• Authors: Sloan Nietert, Ziv Goldfeld, Soroosh Shafiee,
• Abstract summary: We develop an efficient finite-sample algorithm with error bounded by $sqrtvarepsilon k + rho + tildeO(dsqrtkn-1/(k lor 2))$ when $P$ has bounded covariance. Our efficient procedure relies on a novel trace norm approximation of an ideal yet intractable 2-Wasserstein projection estimator.
• Score: 17.22168727622332
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider learning in an adversarial environment, where an $\varepsilon$-fraction of samples from a distribution $P$ are arbitrarily modified (global corruptions) and the remaining perturbations have average magnitude bounded by $\rho$ (local corruptions). Given access to $n$ such corrupted samples, we seek a computationally efficient estimator $\hat{P}_n$ that minimizes the Wasserstein distance $\mathsf{W}_1(\hat{P}_n,P)$. In fact, we attack the fine-grained task of minimizing $\mathsf{W}_1(\Pi_\# \hat{P}_n, \Pi_\# P)$ for all orthogonal projections $\Pi \in \mathbb{R}^{d \times d}$, with performance scaling with $\mathrm{rank}(\Pi) = k$. This allows us to account simultaneously for mean estimation ($k=1$), distribution estimation ($k=d$), as well as the settings interpolating between these two extremes. We characterize the optimal population-limit risk for this task and then develop an efficient finite-sample algorithm with error bounded by $\sqrt{\varepsilon k} + \rho + \tilde{O}(d\sqrt{k}n^{-1/(k \lor 2)})$ when $P$ has bounded covariance. This guarantee holds uniformly in $k$ and is minimax optimal up to the sub-optimality of the plug-in estimator when $\rho = \varepsilon = 0$. Our efficient procedure relies on a novel trace norm approximation of an ideal yet intractable 2-Wasserstein projection estimator. We apply this algorithm to robust stochastic optimization, and, in the process, uncover a new method for overcoming the curse of dimensionality in Wasserstein distributionally robust optimization.

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