Optimal Randomized Group Testing Algorithm to Determine the Number of Defectives

We study the problem of determining the exact number of defective items in an adaptive group testing by using a minimum number of tests. We improve the existing algorithm and prove a lower bound that shows that the number of tests in our algorithm is optimal up to small additive terms

Improved Generalization Bounds for Robust Learning

We consider a model of robust learning in an adversarial environment. The learner gets uncorrupted training data with access to possible corruptions that may be affected by the adversary during testing. The learner's goal is to build a robust classifier that would be tested on future adversarial examples. We use a zero-sum game between the learner and the adversary as our game theoretic framework. The adversary is limited to $k$ possible corruptions for each input. Our model is closely related to the adversarial examples model of Schmidt et al. (2018); Madry et al. (2017). Our main results consist of generalization bounds for the binary and multi-class classification, as well as the real-valued case (regression). For the binary classification setting, we both tighten the generalization bound of Feige, Mansour, and Schapire (2015), and also are able to handle an infinite hypothesis class $H$. The sample complexity is improved from $O(\frac{1}{\epsilon^4}\log(\frac{|H|}{\delta}))$ to $O(\frac{1}{\epsilon^2}(k\log(k)VC(H)+\log\frac{1}{\delta}))$. Additionally, we extend the algorithm and generalization bound from the binary to the multiclass and real-valued cases. Along the way, we obtain results on fat-shattering dimension and Rademacher complexity of $k$-fold maxima over function classes; these may be of independent interest. For binary classification, the algorithm of Feige et al. (2015) uses a regret minimization algorithm and an ERM oracle as a blackbox; we adapt it for the multi-class and regression settings. The algorithm provides us with near-optimal policies for the players on a given training sample.Comment: Appearing at the 30th International Conference on Algorithmic Learning Theory (ALT 2019

On Detecting Some Defective Items in Group Testing

Group testing is an approach aimed at identifying up to $d$ defective items among a total of $n$ elements. This is accomplished by examining subsets to determine if at least one defective item is present. In our study, we focus on the problem of identifying a subset of $\ell\leq d$ defective items. We develop upper and lower bounds on the number of tests required to detect $\ell$ defective items in both the adaptive and non-adaptive settings while considering scenarios where no prior knowledge of $d$ is available, and situations where an estimate of $d$ or at least some non-trivial upper bound on $d$ is available. When no prior knowledge on $d$ is available, we prove a lower bound of $\Omega(\frac{\ell \log^2n}{\log \ell +\log\log n})$ tests in the randomized non-adaptive settings and an upper bound of $O(\ell \log^2 n)$ for the same settings. Furthermore, we demonstrate that any non-adaptive deterministic algorithm must ask $\Theta(n)$ tests, signifying a fundamental limitation in this scenario. For adaptive algorithms, we establish tight bounds in different scenarios. In the deterministic case, we prove a tight bound of $\Theta(\ell\log{(n/\ell)})$. Moreover, in the randomized settings, we derive a tight bound of $\Theta(\ell\log{(n/d)})$. When $d$, or at least some non-trivial estimate of $d$, is known, we prove a tight bound of $\Theta(d\log (n/d))$ for the deterministic non-adaptive settings, and $\Theta(\ell\log(n/d))$ for the randomized non-adaptive settings. In the adaptive case, we present an upper bound of $O(\ell \log (n/\ell))$ for the deterministic settings, and a lower bound of $\Omega(\ell\log(n/d)+\log n)$. Additionally, we establish a tight bound of $\Theta(\ell \log(n/d))$ for the randomized adaptive settings

Best Arm Identification in Stochastic Bandits: Beyond β−\beta-optimality

This paper investigates a hitherto unaddressed aspect of best arm identification (BAI) in stochastic multi-armed bandits in the fixed-confidence setting. Two key metrics for assessing bandit algorithms are computational efficiency and performance optimality (e.g., in sample complexity). In stochastic BAI literature, there have been advances in designing algorithms to achieve optimal performance, but they are generally computationally expensive to implement (e.g., optimization-based methods). There also exist approaches with high computational efficiency, but they have provable gaps to the optimal performance (e.g., the $\beta$-optimal approaches in top-two methods). This paper introduces a framework and an algorithm for BAI that achieves optimal performance with a computationally efficient set of decision rules. The central process that facilitates this is a routine for sequentially estimating the optimal allocations up to sufficient fidelity. Specifically, these estimates are accurate enough for identifying the best arm (hence, achieving optimality) but not overly accurate to an unnecessary extent that creates excessive computational complexity (hence, maintaining efficiency). Furthermore, the existing relevant literature focuses on the family of exponential distributions. This paper considers a more general setting of any arbitrary family of distributions parameterized by their mean values (under mild regularity conditions). The optimality is established analytically, and numerical evaluations are provided to assess the analytical guarantees and compare the performance with those of the existing ones