Online Discrepancy Minimization for Stochastic Arrivals

In the stochastic online vector balancing problem, vectors $v_1,v_2,\ldots,v_T$ chosen independently from an arbitrary distribution in $\mathbb{R}^n$ arrive one-by-one and must be immediately given a $\pm$ sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm. We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to $\mathsf{polylog}(nT)$ factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC' 20). In particular, for the Koml\'{o}s problem where $\|v_t\|_2\leq 1$ for each $t$, our algorithm achieves $\tilde{O}(1)$ discrepancy with high probability, improving upon the previous $\tilde{O}(n^{3/2})$ bound. For Tusn\'{a}dy's problem of minimizing the discrepancy of axis-aligned boxes, we obtain an $O(\log^{d+4} T)$ bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker $O(\log^{2d+1} T)$ bound. We also consider the Banaszczyk setting, where given a symmetric convex body $K$ with Gaussian measure at least $1/2$, our algorithm achieves $\tilde{O}(1)$ discrepancy with respect to the norm given by $K$ for input distributions with sub-exponential tails. Our key idea is to introduce a potential that also enforces constraints on how the discrepancy vector evolves, allowing us to maintain certain anti-concentration properties. For the Banaszczyk setting, we further enhance this potential by combining it with ideas from generic chaining. Finally, we also extend these results to the setting of online multi-color discrepancy

An evaluation of microleakage of various glass ionomer based restorative materials in deciduous and permanent teeth: An in vitro study

AbstractAimTo evaluate the microleakage of recently available glass ionomer based restorative materials (GC Fuji IX GP, GC Fuji VII, and Dyract) and compare their microleakage with the previously existing glass ionomer restorative materials (GC Fuji II LC) in primary and permanent teeth.MethodOne hundred and fifty (75+75) non-carious deciduous and permanent teeth were restored with glass ionomer based restorative materials after making class I cavities. Samples were subjected to thermocycling after storing in distilled water for 24h. Two coats of nail polish were applied 1mm short of restorative margins and samples sectioned buccolingually after storing in methylene blue dye for 24h. Microleakage was assessed using stereomicroscope.ResultSignificant differences (P<0.05) were found when inter group comparisons were done. Except when GC Fuji VII (Group III) was compared with GC Fuji II LC (Group II) and Dyract (Group IV), non-significant differences (P>0.05) were observed. It was found that there was no statistically significant difference when the means of microleakage of primary teeth were compared with those of permanent teeth.ConclusionsGC Fuji IX GP showed maximum microleakage and GC Fuji VII showed least microleakage

Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing

A well-known result of Banaszczyk in discrepancy theory concerns the prefix discrepancy problem (also known as the signed series problem): given a sequence of $T$ unit vectors in $\mathbb{R}^d$, find $\pm$ signs for each of them such that the signed sum vector along any prefix has a small $\ell_\infty$-norm? This problem is central to proving upper bounds for the Steinitz problem, and the popular Koml\'os problem is a special case where one is only concerned with the final signed sum vector instead of all prefixes. Banaszczyk gave an $O(\sqrt{\log d+ \log T})$ bound for the prefix discrepancy problem. We investigate the tightness of Banaszczyk's bound and consider natural generalizations of prefix discrepancy: We first consider a smoothed analysis setting, where a small amount of additive noise perturbs the input vectors. We show an exponential improvement in $T$ compared to Banaszczyk's bound. Using a primal-dual approach and a careful chaining argument, we show that one can achieve a bound of $O(\sqrt{\log d+ \log\!\log T})$ with high probability in the smoothed setting. Moreover, this smoothed analysis bound is the best possible without further improvement on Banaszczyk's bound in the worst case. We also introduce a generalization of the prefix discrepancy problem where the discrepancy constraints correspond to paths on a DAG on $T$ vertices. We show that an analog of Banaszczyk's $O(\sqrt{\log d+ \log T})$ bound continues to hold in this setting for adversarially given unit vectors and that the $\sqrt{\log T}$ factor is unavoidable for DAGs. We also show that the dependence on $T$ cannot be improved significantly in the smoothed case for DAGs. We conclude by exploring a more general notion of vector balancing, which we call combinatorial vector balancing. We obtain near-optimal bounds in this setting, up to poly-logarithmic factors.Comment: 22 pages. Appear in ITCS 202

Investigating the Application of Transfer Learning Techniques in Cloud-Based AI Systems for Improved Performance and Reduced Training Time

This current research paper examines the adaptive technology solution approaches of transfer learning in a cloud environment for AI systems’ enhanced results and faster training periods. Concerning transfer learning methods, their application with cloud computing environments, and their effects on the efficiency of the AI model are the subject of the study. In this work, after reviewing the current literature and the state of the art of transfer learning and cloud-based AI, we discuss their integration’s prospects and opportunities for scalability, data privacy, and model generalization. The study sheds light on how transfer learning can go a long way in strengthening the efficiency of cloud AI, especially in facets such as speech and language processing, image identification, and speech recognition. The results of our study point out that it is possible to significantly improve the efficiency of model training and accuracy by applying transfer learning methodologies, thus opening opportunities for more dynamic AI solutions in the cloud context

Online discrepancy minimization for stochastic arrivals

In the stochastic online vector balancing problem, vectors v1,v2,…,vT chosen independently from an arbitrary distribution in Rn arrive one-by-one and must be immediately given a ± sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm. We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to polylog(nT) factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC' 20). In particular, for the Komlos problem where ∥v_t∥_2≤1 for each t, our algorithm achieves ˜O(1) discrepancy with high probability, improving upon the previous ˜O(n3/2) bound. For Tusnády's problem of minimizing the discrepancy of axis-aligned boxes, we obtain an O(log^{d+4}T) bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker O(log^{2d+1}T) bound. We also consider the Banaszczyk setting, where given a symmetric convex body K with Gaussian measure at least 1/2, our algorithm achieves \tilde{O}(1) discrepancy with respect to the norm given by K for input distributions with sub-exponential tails. Our results are based on a new potential function approach. Previous techniques consider a potential that penalizes large discrepancy, and greedily chooses the next color to minimize the increase in potential. Our key idea is to introduce a potential that also enforces constraints on how the discrepancy vector evolves, allowing us to maintain certain anti-concentration properties. We believe that our techniques to control the evolution of states could find other applications in stochastic processes and online algorithms. For the Banaszczyk setting, we further enhance this potential by combining it with ideas from generic chaining. Finally, we also extend these results to the setting of online multi-color discrepancy.</p

Online discrepancy minimization for stochastic arrivals

In the stochastic online vector balancing problem, vectors v1, v2,..., vT chosen independently from an arbitrary distribution in Rn arrive one-by-one and must be immediately given a ± sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm. We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to polylog(nT) factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC' 20). In particular, for the Komlós problem where kvtk2 ≤ 1 for each t, our algorithm achieves Oe(1) discrepancy with high probability, improving upon the previous Oe(n3/2) bound. For Tusnády's problem of minimizing the discrepancy of axis-aligned boxes, we obtain an O(logd+4 T) bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker O(log2d+1 T) bound. We also consider the Banaszczyk setting, where given a symmetric convex body K with Gaussian measure at least 1/2, our algorithm achieves Oe(1) discrepancy with respect to the norm given by K for input distributions with sub-exponential tails. Our results are based on a new potential function approach. Previous techniques consider a potential that penalizes large discrepancy, and greedily chooses the next color to minimize the increase in potential. Our key idea is to introduce a potential that also enforces constraints on how the discrepancy vector evolves, allowing us to maintain certain anti-concentration properties. We believe that our techniques to control the evolution of states could find other applications in stochastic processes and online algorithms. For the Banaszczyk setting, we further enhance this potential by combining it with ideas from generic chaining. Finally, we also extend these results to the setting of online multicolor discrepancy

Comparative evaluation of oxytetracycline versus QMix, MTAD, and ethylenediaminetetraacetic acid as smear layer removal agents: An in vitro study

Aim: The aim is to evaluate the smear layer removal efficacy of Oxytetracycline and compare it with different endodontic irrigating solutions (QMix, mixture of tetracycline, citric acid, and detergent [MTAD] and ethylenediaminetetraacetic acid [EDTA]) under scanning electron microscope (SEM). Materials and Methods: Fifty single-rooted teeth were decoronated. All the root canals were shaped with K3XF rotary Ni-Ti instruments (#30.,06) and irrigation was done with 2 ml of 5.25% sodium hypochlorite (NaOCl) solution between each file change. The samples were divided randomly into the 5 groups (n = 10) depending on the final irrigant used as follows: 5.25% NaOCl as control (Group 1), 17% EDTA (Group 2), QMix (Group 3), BioPure MTAD (Group 4), and Oxytetracycline (Group 5). The final irrigation was done using 5 ml of the respective irrigating solutions, each for 60 s. Specimens were fractured longitudinally. For each root, the half containing most visible part of apex was selected for the SEM analysis. The presence/absence of smear layer was evaluated at the coronal, middle, and apical third of the root canal using three score criteria. Data were analyzed using Mann–Whitney U-test and Kruskal–Wallis test. Results: QMix showed the least smear layer scores followed by MTAD, EDTA, NaOCl, and Oxytetracycline. Conclusion: Within the limitation of this study, it can be concluded that the test irrigant Oxytetracycline failed to remove the smear layer throughout the entire length of the root canal