Cheng Chen

Jize Xie, Cheng Chen, Zhiyong Wang et al.

Online learning to rank sequentially recommends a small list of items to users from a large candidate set and receives the users' click feedback. In many real-world scenarios, users browse the recommended list in order and click the first attractive item without checking the rest. Such behaviors are usually formulated as the cascade model. Many recent works study algorithms for cascading bandits, an online learning to rank framework in the cascade model. However, the performance of existing methods may drop significantly if part of the user feedback is adversarially corrupted (e.g., click fraud). In this work, we study how to resist adversarial corruptions in cascading bandits. We first formulate the ``\textit{Cascading Bandits with Adversarial Corruptions}" (CBAC) problem, which assumes that there is an adaptive adversary that may manipulate the user feedback. Then we propose two robust algorithms for this problem, which assume the corruption level is known and agnostic, respectively. We show that both algorithms can achieve logarithmic regret when the algorithm is not under attack, and the regret increases linearly with the corruption level. The experimental results also verify the robustness of our methods.

Chen Cheng, Yuting Wei, Yuxin Chen

This paper aims to address two fundamental challenges arising in eigenvector estimation and inference for a low-rank matrix from noisy observations: (1) how to estimate an unknown eigenvector when the eigen-gap (i.e. the spacing between the associated eigenvalue and the rest of the spectrum) is particularly small; (2) how to perform estimation and inference on linear functionals of an eigenvector -- a sort of "fine-grained" statistical reasoning that goes far beyond the usual $\ell_2$ analysis. We investigate how to address these challenges in a setting where the unknown $n\times n$ matrix is symmetric and the additive noise matrix contains independent (and non-symmetric) entries. Based on eigen-decomposition of the asymmetric data matrix, we propose estimation and uncertainty quantification procedures for an unknown eigenvector, which further allow us to reason about linear functionals of an unknown eigenvector. The proposed procedures and the accompanying theory enjoy several important features: (1) distribution-free (i.e. prior knowledge about the noise distributions is not needed); (2) adaptive to heteroscedastic noise; (3) minimax optimal under Gaussian noise. Along the way, we establish optimal procedures to construct confidence intervals for the unknown eigenvalues. All this is guaranteed even in the presence of a small eigen-gap (up to $O(\sqrt{n/\mathrm{poly}\log (n)})$ times smaller than the requirement in prior theory), which goes significantly beyond what generic matrix perturbation theory has to offer.

Yuxin Chen, Chen Cheng, Jianqing Fan

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly perturbed version $\mathbf{M}$ is observed. The noise matrix $\mathbf{M}-\mathbf{M}^{\star}$ is composed of zero-mean independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise, for example, when we have two independent samples for each entry of $\mathbf{M}^{\star}$ and arrange them into an {\em asymmetric} data matrix $\mathbf{M}$. The aim is to estimate the leading eigenvalue and eigenvector of $\mathbf{M}^{\star}$. We demonstrate that the leading eigenvalue of the data matrix $\mathbf{M}$ can be $O(\sqrt{n})$ times more accurate --- up to some log factor --- than its (unadjusted) leading singular value in eigenvalue estimation. Further, the perturbation of any linear form of the leading eigenvector of $\mathbf{M}$ --- say, entrywise eigenvector perturbation --- is provably well-controlled. This eigen-decomposition approach is fully adaptive to heteroscedasticity of noise without the need of careful bias correction or any prior knowledge about the noise variance. We also provide partial theory for the more general rank-$r$ case. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigen-decomposition could sometimes be beneficial.