Adversarially-Trained Nonnegative Matrix Factorization

We consider an adversarially-trained version of the nonnegative matrix factorization, a popular latent dimensionality reduction technique. In our formulation, an attacker adds an arbitrary matrix of bounded norm to the given data matrix. We design efficient algorithms inspired by adversarial training to optimize for dictionary and coefficient matrices with enhanced generalization abilities. Extensive simulations on synthetic and benchmark datasets demonstrate the superior predictive performance on matrix completion tasks of our proposed method compared to state-of-the-art competitors, including other variants of adversarial nonnegative matrix factorization.

CIFS: Improving Adversarial Robustness of CNNs via Channel-wise Importance-based Feature Selection

We investigate the adversarial robustness of CNNs from the perspective of channel-wise activations. By comparing \textit{non-robust} (normally trained) and \textit{robustified} (adversarially trained) models, we observe that adversarial training (AT) robustifies CNNs by aligning the channel-wise activations of adversarial data with those of their natural counterparts. However, the channels that are \textit{negatively-relevant} (NR) to predictions are still over-activated when processing adversarial data. Besides, we also observe that AT does not result in similar robustness for all classes. For the robust classes, channels with larger activation magnitudes are usually more \textit{positively-relevant} (PR) to predictions, but this alignment does not hold for the non-robust classes. Given these observations, we hypothesize that suppressing NR channels and aligning PR ones with their relevances further enhances the robustness of CNNs under AT. To examine this hypothesis, we introduce a novel mechanism, i.e., \underline{C}hannel-wise \underline{I}mportance-based \underline{F}eature \underline{S}election (CIFS). The CIFS manipulates channels' activations of certain layers by generating non-negative multipliers to these channels based on their relevances to predictions. Extensive experiments on benchmark datasets including CIFAR10 and SVHN clearly verify the hypothesis and CIFS's effectiveness of robustifying CNNs.

SGA: A Robust Algorithm for Partial Recovery of Tree-Structured Graphical Models with Noisy Samples

We consider learning Ising tree models when the observations from the nodes are corrupted by independent but non-identically distributed noise with unknown statistics. Katiyar et al. (2020) showed that although the exact tree structure cannot be recovered, one can recover a partial tree structure; that is, a structure belonging to the equivalence class containing the true tree. This paper presents a systematic improvement of Katiyar et al. (2020). First, we present a novel impossibility result by deriving a bound on the necessary number of samples for partial recovery. Second, we derive a significantly improved sample complexity result in which the dependence on the minimum correlation $\rho_{\min}$ is $\rho_{\min}^{-8}$ instead of $\rho_{\min}^{-24}$. Finally, we propose Symmetrized Geometric Averaging (SGA), a more statistically robust algorithm for partial tree recovery. We provide error exponent analyses and extensive numerical results on a variety of trees to show that the sample complexity of SGA is significantly better than the algorithm of Katiyar et al. (2020). SGA can be readily extended to Gaussian models and is shown via numerical experiments to be similarly superior.

Risk-Constrained Thompson Sampling for CVaR Bandits

The multi-armed bandit (MAB) problem is a ubiquitous decision-making problem that exemplifies the exploration-exploitation tradeoff. Standard formulations exclude risk in decision making. Risk notably complicates the basic reward-maximising objective, in part because there is no universally agreed definition of it. In this paper, we consider a popular risk measure in quantitative finance known as the Conditional Value at Risk (CVaR). We explore the performance of a Thompson Sampling-based algorithm CVaR-TS under this risk measure. We provide comprehensive comparisons between our regret bounds with state-of-the-art L/UCB-based algorithms in comparable settings and demonstrate their clear improvement in performance. We also include numerical simulations to empirically verify that CVaR-TS outperforms other L/UCB-based algorithms.

Probabilistic Sequential Shrinking: A Best Arm Identification Algorithm for Stochastic Bandits with Corruptions

We consider a best arm identification (BAI) problem for stochastic bandits with adversarial corruptions in the fixed-budget setting of $T$ steps. We design a novel randomized algorithm, Probabilistic Sequential Shrinking$(u)$ (PSS$(u)$), which is agnostic to the amount of corruptions. When the amount of corruptions per step (CPS) is below a threshold, PSS$(u)$ identifies the best arm or item with probability tending to $1$ as $T\rightarrow\infty$. Otherwise, the optimality gap of the identified item degrades gracefully with the CPS. We argue that such a bifurcation is necessary. In addition, we show that when the CPS is sufficiently large, no algorithm can achieve a BAI probability tending to $1$ as $T\rightarrow \infty$. In PSS$(u)$, the parameter $u$ serves to balance between the optimality gap and success probability. En route, the injection of randomization is shown to be essential to mitigate the impact of corruptions. Indeed, we show that PSS$(u)$ has a better performance than its deterministic analogue, the Successive Halving (SH) algorithm by Karnin et al. (2013). PSS$(2)$'s performance guarantee matches SH's when there is no corruption. Finally, we identify a term in the exponent of the failure probability of PSS$(u)$ that generalizes the common $H_2$ term for BAI under the fixed-budget setting.

Positive Semidefinite Matrix Factorization: A Connection with Phase Retrieval and Affine Rank Minimization

Positive semidefinite matrix factorization (PSDMF) expresses each entry of a nonnegative matrix as the inner product of two positive semidefinite (psd) matrices. When all these psd matrices are constrained to be diagonal, this model is equivalent to nonnegative matrix factorization. Applications include combinatorial optimization, quantum-based statistical models, and recommender systems, among others. However, despite the increasing interest in PSDMF, only a few PSDMF algorithms were proposed in the literature. In this paper, we show that PSDMF algorithms can be designed based on phase retrieval (PR) and affine rank minimization (ARM) algorithms. This procedure allows a significant shortcut in designing new PSDMF algorithms, as it allows to leverage some of the useful numerical properties of existing PR and ARM methods to the PSDMF framework. Motivated by this idea, we introduce a new family of PSDMF algorithms based on singular value projection (SVP) and iterative hard thresholding (IHT). This family subsumes previously-proposed projected gradient PSDMF methods; additionally, we show a new connection between SVP-based methods and majorization-minimization. Numerical experiments show that our proposed methods outperform state-of-the-art coordinate descent algorithms in terms of convergence speed and computational complexity, in certain scenarios. In certain cases, our proposed normalized-IHT-based method is the only algorithm able to find a solution. These results support our claim that the PSDMF framework can inherit desired numerical properties from PR and ARM algorithms, leading to more efficient PSDMF algorithms, and motivate further study of the links between these models.

MC2G: An Efficient Algorithm for Matrix Completion with Social and Item Similarity Graphs

We consider a discrete-valued matrix completion problem for recommender systems in which both the social and item similarity graphs are available as side information. We develop and analyze MC2G (Matrix Completion with 2 Graphs), a quasilinear-time algorithm which is based on spectral clustering and local refinement steps. We show that the sample complexity of MC2G meets an information-theoretic limit that is derived using maximum likelihood estimation and is also order-optimal. We demonstrate that having both graphs as side information outperforms having just a single graph, thus the availability of two graphs results in a synergistic effect. Experiments on synthetic datasets corroborate our theoretical results. Finally, experiments on a sub-sampled version of the Netflix dataset show that MC2G significantly outperforms other state-of-the-art matrix completion algorithms that leverage graph side information.

Exact Asymptotics for Learning Tree-Structured Graphical Models with Side Information: Noiseless and Noisy Samples

Given side information that an Ising tree-structured graphical model is homogeneous and has no external field, we derive the exact asymptotics of learning its structure from independently drawn samples. Our results, which leverage the use of probabilistic tools from the theory of strong large deviations, refine the large deviation (error exponents) results of Tan, Anandkumar, Tong, and Willsky [IEEE Trans. on Inform. Th., 57(3):1714--1735, 2011] and strictly improve those of Bresler and Karzand [Ann. Statist., 2020]. In addition, we extend our results to the scenario in which the samples are observed in random noise. In this case, we show that they strictly improve on the recent results of Nikolakakis, Kalogerias, and Sarwate [Proc. AISTATS, 1771--1782, 2019]. Our theoretical results demonstrate keen agreement with experimental results for sample sizes as small as that in the hundreds.

Optimal Resolution of Change-Point Detection with Empirically Observed Statistics and Erasures

This paper revisits the offline change-point detection problem from a statistical learning perspective. Instead of assuming that the underlying pre- and post-change distributions are known, it is assumed that we have partial knowledge of these distributions based on empirically observed statistics in the form of training sequences. Our problem formulation finds a variety of real-life applications from detecting when climate change occurred to detecting when a virus mutated. Using the training sequences as well as the test sequence consisting of a single-change and allowing for the erasure or rejection option, we derive the optimal resolution between the estimated and true change-points under two different asymptotic regimes on the undetected error probability---namely, the large and moderate deviations regimes. In both regimes, strong converses are also proved. In the moderate deviations case, the optimal resolution is a simple function of a symmetrized version of the chi-square distance.