Many real-world problems can often be cast as the optimization of DR-submodular functions defined over a convex domain. These functions play an important role with applications in many areas of applied mathematics, such as machine learning, computer vision, operation research, communication systems or economics. In addition, they capture a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we show that for maximizing non-monotone DR-submodular functions over a general convex set (such as up-closed convex sets, conic convex set, etc) the Frank-Wolfe algorithm achieves an approximation guarantee which depends on the convex set. To the best of our knowledge, this is the first approximation guarantee. Finally we benchmark our algorithm on problems arising in machine learning domain with the real-world datasets.

Diminishing-returns (DR) submodular optimization is an important field with many real-world applications in machine learning, economics and communication systems. It captures a subclass of non-convex optimization that provides both practical and theoretical guarantees.In this paper, we study the fundamental problem of maximizing non-monotone DR-submodular functions over down-closed and general convex sets in both offline and online settings. First, we show that for offline maximizing non-monotone DR-submodular functions over a general convex set, the Frank-Wolfe algorithm achieves an approximation guarantee which depends on the convex set. Next, we show that the Stochastic Gradient Ascent algorithm achieves a 1/4-approximation ratio with the regret of O(1/ √ T ) for the problem of maximizing non-monotone DR-submodular functions over down-closed convex sets. These are the first approximation guarantees in the corresponding settings. Finally we benchmark these algorithms on problems arising in machine learning domain with the real-world datasets.

Objective: The ASCO/CAP guidelines for reporting HER2 in breast cancer, first released in 2007, aimed to standardize the reporting protocol, and were updated in 2013 and 2018, to ensure right treatment. Several studies have analyzed the changes attributed to 2013 updated guidelines, and majority of them found increase in positive and equivocal cases. However, the precise implication of these updated guidelines is still contentious, in spite of the latest update (2018 guidelines) addressing some of the issues. We conducted systematic review and meta- analysis to see the impact of 2013 guidelines on various HER2 reporting categories by both FISH and IHC. Materials and Methods: After extensively searching the pertinent literature, 16 studies were included for the systematic review. We divided our approach in three strategies: (1) Studies in which breast cancer cases were scored for HER2 by FISH or IHC as a primary test concurrently by both 2007 and 2013 guidelines, (2) Studies in which HER2 results were equivocal by IHC and were followed by reflex-FISH test by both 2007 and 2013 guidelines, and (3) Studies in which trends of HER2 reporting were compared in the two periods before and after implementation of updated 2013 guidelines. All the paired data in these respective categories was pooled and analyzed statistically to see the overall impact of the updated guidelines. Results: In the first category, by pooled analysis of primary FISH testing there has been a significant increase in the equivocal cases (P < 0.001) and positive cases (P = 0.037). We also found 8.3% and 0.8% of all the negative cases from 2007 guidelines shifted to equivocal and positive categories, respectively. Similarly by primary IHC testing there has been a significant increase in both equivocal cases (P < 0.001) and positive cases (P = 0.02). In the second category of reflex-FISH testing there was a substantial increase in the equivocal cases (P < 0.0001); however there is insignificant decrease (10% to 9.7%; P = 0.66) in the amplified cases. In the third approach for evaluating the trend, with the implementation of 2013 guidelines, there was increase in the equivocal category (P = 0.025) and positive category (P = 0.0088) by IHC. By FISH test also there was significant increase in the equivocal category (P < 0.001) while the increase in the positive category was non-significant (P = 0.159). Conclusions: The updated 2013 guidelines has significantly increased the positive and equivocal cases using primary FISH or IHC test and with further reflex testing, thereby increasing the double equivocal cases and increasing the cost and delaying the decision for definite management. However, whether the additional patients becoming eligible for HDT will derive treatment benefit needs to be answered by further large clinical trials.

When a computer system schedules jobs there is typically a significant cost associated with preempting a job during execution. This cost can be incurred from the expensive task of saving the memory’s state or from loading data into and out of memory. Thus, it is desirable to schedule jobs non-preemptively to avoid the costs of preemption. There is a need for non-preemptive system schedulers for desktops, servers, and data centers. Despite this need, there is a gap between theory and practice. Indeed, few non-preemptive online schedulers are known to have strong theoretical guarantees. This gap is likely due to strong lower bounds on any online algorithm for popular objectives. Indeed, typical worst-case analysis approaches, and even resource-augmented approaches such as speed augmentation, result in all algorithms having poor performance guarantees. This article considers online non-preemptive scheduling problems in the worst-case rejection model where the algorithm is allowed to reject a small fraction of jobs. By rejecting only a few jobs, this article shows that the strong lower bounds can be circumvented. This approach can be used to discover algorithmic scheduling policies with desirable worst-case guarantees. Specifically, the article presents algorithms for the following three objectives: minimizing the total flow-time, minimizing the total weighted flow-time plus energy where energy is a convex function, and minimizing the total energy under the deadline constraints. The algorithms for the first two problems have a small constant competitive ratio while rejecting only a constant fraction of jobs. For the last problem, we present a constant competitive ratio without rejection. Beyond specific results, the article asserts that alternative models beyond speed augmentation should be explored to aid in the discovery of good schedulers in the face of the requirement of being online and non-preemptive.

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