In stochastic low-rank matrix bandit, the expected reward of an arm is equal to the inner product between its feature matrix and some unknown $d_1$ by $d_2$ low-rank parameter matrix $\Theta^*$ with rank $r \ll d_1\wedge d_2$. While all prior studies assume the payoffs are mixed with sub-Gaussian noises, in this work we loosen this strict assumption and consider the new problem of \underline{low}-rank matrix bandit with \underline{h}eavy-\underline{t}ailed \underline{r}ewards (LowHTR), where the rewards only have finite $(1+\delta)$ moment for some $\delta \in (0,1]$. By utilizing the truncation on observed payoffs and the dynamic exploration, we propose a novel algorithm called LOTUS attaining the regret bound of order $\tilde O(d^\frac{3}{2}r^\frac{1}{2}T^\frac{1}{1+\delta}/\tilde{D}_{rr})$ without knowing $T$, which matches the state-of-the-art regret bound under sub-Gaussian noises~\citep{lu2021low,kang2022efficient} with $\delta = 1$. Moreover, we establish a lower bound of the order $\Omega(d^\frac{\delta}{1+\delta} r^\frac{\delta}{1+\delta} T^\frac{1}{1+\delta}) = \Omega(T^\frac{1}{1+\delta})$ for LowHTR, which indicates our LOTUS is nearly optimal in the order of $T$. In addition, we improve LOTUS so that it does not require knowledge of the rank $r$ with $\tilde O(dr^\frac{3}{2}T^\frac{1+\delta}{1+2\delta})$ regret bound, and it is efficient under the high-dimensional scenario. We also conduct simulations to demonstrate the practical superiority of our algorithm.

In the stochastic contextual low-rank matrix bandit problem, the expected reward of an action is given by the inner product between the action's feature matrix and some fixed, but initially unknown $d_1$ by $d_2$ matrix $\Theta^*$ with rank $r \ll \{d_1, d_2\}$, and an agent sequentially takes actions based on past experience to maximize the cumulative reward. In this paper, we study the generalized low-rank matrix bandit problem, which has been recently proposed in \cite{lu2021low} under the Generalized Linear Model (GLM) framework. To overcome the computational infeasibility and theoretical restrain of existing algorithms on this problem, we first propose the G-ESTT framework that modifies the idea from \cite{jun2019bilinear} by using Stein's method on the subspace estimation and then leverage the estimated subspaces via a regularization idea. Furthermore, we remarkably improve the efficiency of G-ESTT by using a novel exclusion idea on the estimated subspace instead, and propose the G-ESTS framework. We also show that G-ESTT can achieve the $\tilde{O}(\sqrt{(d_1+d_2)MrT})$ bound of regret while G-ESTS can achineve the $\tilde{O}(\sqrt{(d_1+d_2)^{3/2}Mr^{3/2}T})$ bound of regret under mild assumption up to logarithm terms, where $M$ is some problem dependent value. Under a reasonable assumption that $M = O((d_1+d_2)^2)$ in our problem setting, the regret of G-ESTT is consistent with the current best regret of $\tilde{O}((d_1+d_2)^{3/2} \sqrt{rT}/D_{rr})$~\citep{lu2021low} ($D_{rr}$ will be defined later). For completeness, we conduct experiments to illustrate that our proposed algorithms, especially G-ESTS, are also computationally tractable and consistently outperform other state-of-the-art (generalized) linear matrix bandit methods based on a suite of simulations.

Multi-armed Bandit (MAB) [3] is a fundamental and powerful framework in sequential decisionmaking problems. Given the potential existence of malicious users in real-world scenarios [10], a recent line of works considers the stochastic bandit problem under adversarial corruptions: an agent adaptively updates its policy to choose an arm from the arm set, and an adversary may contaminate the reward generated from the stochastic bandit before the agent could observe it. To robustify bandit learning algorithms under adversarial corruptions, several algorithms have been developed in the setting of traditional MAB [16, 18, 29] and contextual linear bandits [6, 12, 17, 27, 39]. These works consider either the weak adversary [29], which has access to all past data but not the current action before choosing its attack, or the strong adversary [6], which is also aware of the current action for contamination. Details of these two adversaries will be elaborated in Section 3. In practice, bandits under adversarial corruptions can be used in many real-world problems such as pay-per-click advertising with click fraud and recommendation systems with fake reviews [29], and it has been empirically validated that stochastic MABs are vulnerable to slight corruption [12, 15, 18]. Although there has been extensive research on the adversarial robustness of stochastic bandits, most existing works consider problems with a discrete arm set, such as the traditional MAB and contextual linear bandit. In this paper, we investigate robust bandit algorithms against adversarial corruptions in the Lipschitz bandit setting, where a continuously infinite arm set lie in a known metric space with covering dimension d and the expected reward function is an unknown Lipschitz function.

The rapid growth of deep learning (DL) has spurred interest in enhancing log-based anomaly detection. This approach aims to extract meaning from log events (log message templates) and develop advanced DL models for anomaly detection. However, these DL methods face challenges like heavy reliance on training data, labels, and computational resources due to model complexity. In contrast, traditional machine learning and data mining techniques are less data-dependent and more efficient but less effective than DL. To make log-based anomaly detection more practical, the goal is to enhance traditional techniques to match DL's effectiveness. Previous research in a different domain (linking questions on Stack Overflow) suggests that optimized traditional techniques can rival state-of-the-art DL methods. Drawing inspiration from this concept, we conducted an empirical study. We optimized the unsupervised PCA (Principal Component Analysis), a traditional technique, by incorporating lightweight semantic-based log representation. This addresses the issue of unseen log events in training data, enhancing log representation. Our study compared seven log-based anomaly detection methods, including four DL-based, two traditional, and the optimized PCA technique, using public and industrial datasets. Results indicate that the optimized unsupervised PCA technique achieves similar effectiveness to advanced supervised/semi-supervised DL methods while being more stable with limited training data and resource-efficient. This demonstrates the adaptability and strength of traditional techniques through small yet impactful adaptations.

In stochastic contextual bandits, an agent sequentially makes actions from a time-dependent action set based on past experience to minimize the cumulative regret. Like many other machine learning algorithms, the performance of bandits heavily depends on their multiple hyperparameters, and theoretically derived parameter values may lead to unsatisfactory results in practice. Moreover, it is infeasible to use offline tuning methods like cross-validation to choose hyperparameters under the bandit environment, as the decisions should be made in real time. To address this challenge, we propose the first online continuous hyperparameter tuning framework for contextual bandits to learn the optimal parameter configuration within a search space on the fly. Specifically, we use a double-layer bandit framework named CDT (Continuous Dynamic Tuning) and formulate the hyperparameter optimization as a non-stationary continuum-armed bandit, where each arm represents a combination of hyperparameters, and the corresponding reward is the algorithmic result. For the top layer, we propose the Zooming TS algorithm that utilizes Thompson Sampling (TS) for exploration and a restart technique to get around the switching environment. The proposed CDT framework can be easily used to tune contextual bandit algorithms without any pre-specified candidate set for hyperparameters. We further show that it could achieve sublinear regret in theory and performs consistently better on both synthetic and real datasets in practice.

Adaptive gradient methods, such as \textsc{Adam}, have achieved tremendous success in machine learning. Scaling gradients by square roots of the running averages of squared past gradients, such methods are able to attain rapid training of modern deep neural networks. Nevertheless, they are observed to generalize worse than stochastic gradient descent (\textsc{SGD}) and tend to be trapped in local minima at an early stage during training. Intriguingly, we discover that substituting the gradient in the preconditioner term with the momentumized version in \textsc{Adam} can well solve the issues. The intuition is that gradient with momentum contains more accurate directional information and therefore its second moment estimation is a better choice for scaling than raw gradient's. Thereby we propose \textsc{AdaMomentum} as a new optimizer reaching the goal of training faster while generalizing better. We further develop a theory to back up the improvement in optimization and generalization and provide convergence guarantee under both convex and nonconvex settings. Extensive experiments on various models and tasks demonstrate that \textsc{AdaMomentum} exhibits comparable performance to \textsc{SGD} on vision tasks, and achieves state-of-the-art results consistently on other tasks including language processing.