Finding Optimal Tangent Points for Reducing Distortions of Hard-label Attacks

One major problem in black-box adversarial attacks is the high query complexity in the hard-label attack setting, where only the top-1 predicted label is available. In this paper, we propose a novel geometric-based approach called Tangent Attack (TA), which identifies an optimal tangent point of a virtual hemisphere located on the decision boundary to reduce the distortion of the attack. Assuming the decision boundary is locally flat, we theoretically prove that the minimum $\ell_2$ distortion can be obtained by reaching the decision boundary along the tangent line passing through such tangent point in each iteration. To improve the robustness of our method, we further propose a generalized method which replaces the hemisphere with a semi-ellipsoid to adapt to curved decision boundaries. Our approach is free of hyperparameters and pre-training. Extensive experiments conducted on the ImageNet and CIFAR-10 datasets demonstrate that our approach can consume only a small number of queries to achieve the low-magnitude distortion. The implementation source code is released online at https://github.com/machanic/TangentAttack.

Robust High Dimensional Expectation Maximization Algorithm via Trimmed Hard Thresholding

In this paper, we study the problem of estimating latent variable models with arbitrarily corrupted samples in high dimensional space ({\em i.e.,} $d\gg n$) where the underlying parameter is assumed to be sparse. Specifically, we propose a method called Trimmed (Gradient) Expectation Maximization which adds a trimming gradients step and a hard thresholding step to the Expectation step (E-step) and the Maximization step (M-step), respectively. We show that under some mild assumptions and with an appropriate initialization, the algorithm is corruption-proofing and converges to the (near) optimal statistical rate geometrically when the fraction of the corrupted samples $\epsilon$ is bounded by $ \tilde{O}(\frac{1}{\sqrt{n}})$. Moreover, we apply our general framework to three canonical models: mixture of Gaussians, mixture of regressions and linear regression with missing covariates. Our theory is supported by thorough numerical results.

Estimating Stochastic Linear Combination of Non-linear Regressions Efficiently and Scalably

Recently, many machine learning and statistical models such as non-linear regressions, the Single Index, Multi-index, Varying Coefficient Index Models and Two-layer Neural Networks can be reduced to or be seen as a special case of a new model which is called the \textit{Stochastic Linear Combination of Non-linear Regressions} model. However, due to the high non-convexity of the problem, there is no previous work study how to estimate the model. In this paper, we provide the first study on how to estimate the model efficiently and scalably. Specifically, we first show that with some mild assumptions, if the variate vector $x$ is multivariate Gaussian, then there is an algorithm whose output vectors have $\ell_2$-norm estimation errors of $O(\sqrt{\frac{p}{n}})$ with high probability, where $p$ is the dimension of $x$ and $n$ is the number of samples. The key idea of the proof is based on an observation motived by the Stein's lemma. Then we extend our result to the case where $x$ is bounded and sub-Gaussian using the zero-bias transformation, which could be seen as a generalization of the classic Stein's lemma. We also show that with some additional assumptions there is an algorithm whose output vectors have $\ell_\infty$-norm estimation errors of $O(\frac{1}{\sqrt{p}}+\sqrt{\frac{p}{n}})$ with high probability. We also provide a concrete example to show that there exists some link function which satisfies the previous assumptions. Finally, for both Gaussian and sub-Gaussian cases we propose a faster sub-sampling based algorithm and show that when the sub-sample sizes are large enough then the estimation errors will not be sacrificed by too much. Experiments for both cases support our theoretical results. To the best of our knowledge, this is the first work that studies and provides theoretical guarantees for the stochastic linear combination of non-linear regressions model.

Consistent $k$-Median: Simpler, Better and Robust

In this paper we introduce and study the online consistent $k$-clustering with outliers problem, generalizing the non-outlier version of the problem studied in [Lattanzi-Vassilvitskii, ICML17]. We show that a simple local-search based online algorithm can give a bicriteria constant approximation for the problem with $O(k^2 \log^2 (nD))$ swaps of medians (recourse) in total, where $D$ is the diameter of the metric. When restricted to the problem without outliers, our algorithm is simpler, deterministic and gives better approximation ratio and recourse, compared to that of [Lattanzi-Vassilvitskii, ICML17].

Distributed $k$-Clustering for Data with Heavy Noise

In this paper, we consider the $k$-center/median/means clustering with outliers problems (or the $(k, z)$-center/median/means problems) in the distributed setting. Most previous distributed algorithms have their communication costs linearly depending on $z$, the number of outliers. Recently Guha et al. overcame this dependence issue by considering bi-criteria approximation algorithms that output solutions with $2z$ outliers. For the case where $z$ is large, the extra $z$ outliers discarded by the algorithms might be too large, considering that the data gathering process might be costly. In this paper, we improve the number of outliers to the best possible $(1+\epsilon)z$, while maintaining the $O(1)$-approximation ratio and independence of communication cost on $z$. The problems we consider include the $(k, z)$-center problem, and $(k, z)$-median/means problems in Euclidean metrics. Implementation of the our algorithm for $(k, z)$-center shows that it outperforms many previous algorithms, both in terms of the communication cost and quality of the output solution.