Finding an approximate second-order stationary point (SOSP) is a well-studied and fundamental problem in stochastic nonconvex optimization with many applications in machine learning. However, this problem is poorly understood in the presence of outliers, limiting the use of existing nonconvex algorithms in adversarial settings. In this paper, we study the problem of finding SOSPs in the strong contamination model, where a constant fraction of datapoints are arbitrarily corrupted. We introduce a general framework for efficiently finding an approximate SOSP with dimension-independent accuracy guarantees, using $$\widetilde{O}({D^2}/{\epsilon})$$ samples where $$D$$ is the ambient dimension and $$\epsilon$$ is the fraction of corrupted datapoints. As a concrete application of our framework, we apply it to the problem of low rank matrix sensing, developing efficient and provably robust algorithms that can tolerate corruptions in both the sensing matrices and the measurements. In addition, we establish a Statistical Query lower bound providing evidence that the quadratic dependence on $$D$$ in the sample complexity is necessary for computationally efficient algorithms.
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Influence Diagnostics under Self-concordance
Influence diagnostics such as influence functions and approximate maximum influence perturbations are popular in machine learning and in AI domain applications. Influence diagnostics are powerful statistical tools to identify influential datapoints or subsets of datapoints. We establish finite-sample statistical bounds, as well as computational complexity bounds, for influence functions and approximate maximum influence perturbations using efficient inverse-Hessian-vector product implementations. We illustrate our results with generalized linear models and large attention based models on synthetic and real data.
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- Award ID(s):
- 2134012
- PAR ID:
- 10444882
- Editor(s):
- Ruiz, Francisco; Dy, Jennifer; van de Meent, Jan-Willem
- Date Published:
- Journal Name:
- Proceedings of The 26th International Conference on Artificial Intelligence and Statistics
- Volume:
- 206
- Page Range / eLocation ID:
- 10028--10076
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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