Global Convergence of SGD On Two Layer Neural Nets

In this note we demonstrate provable convergence of SGD to the global minima of appropriately regularized $\ell_2-$empirical risk of depth $2$ nets -- for arbitrary data and with any number of gates, if they are using adequately smooth and bounded activations like sigmoid and tanh. We build on the results in [1] and leverage a constant amount of Frobenius norm regularization on the weights, along with sampling of the initial weights from an appropriate distribution. We also give a continuous time SGD convergence result that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence loss functions on constant sized neural nets which are "Villani Functions". [1] Bin Shi, Weijie J. Su, and Michael I. Jordan. On learning rates and schr\"odinger operators, 2020. arXiv:2004.06977Comment: 23 pages, 6 figures. Extended abstract accepted at DeepMath 2022. v2 update: New experiments added in Section 3.2 to study the effect of the regularization value. Statement of Theorem 3.4 about SoftPlus nets has been improve

Optimizing Empiric Vancomycin Use in Febrile Neutropenia Patients

Introduction: Febrile neutropenia (FN) is a complication of chemotherapy resulting in a temperature 100.4⁰F or greater plus an absolute neutrophil count (ANC) below 500 cells/mm3 or an ANC below 1000 cells/mm3 and expected to decrease below 500 cells/mm3 within 48 hours. Timely administration of broad-spectrum antimicrobial therapy is a cornerstone for the initial management of FN. However, prolonged empiric antimicrobial treatment can lead to resistance and toxicity. National guidelines and published literature do not support vancomycin as a routine part of empiric antimicrobial regimens in FN; it is only recommended in patients with specific clinical indications. This study aims to evaluate current use of vancomycin in FN and improve compliance with a guideline-driven algorithm (GDA) to ensure appropriate prescribing and therapy duration in a community hospital. Methods: This single-center, biphasic, quality improvement project includes both retrospective (Phase 1) and prospective (Phase II) review of adult patients receiving empiric vancomycin therapy for FN at Baptist Hospital. Phase I took place from January to December 2019, while Phase II took place from January to March 2021. Data collection points included patient demographics, indication, pertinent labs and cultures, therapy duration and compliance with the GDA. In Phase II, pharmacist interventions were made to recommend de-escalation or discontinuation of vancomycin as warranted. Primary outcomes included percent compliance with the GDA and duration of vancomycin therapy, while secondary outcomes included number of pharmacy interventions made and accepted in Phase II. Results: A total of 48 patients were reviewed during phase I of the study while 32 patients were reviewed during phase II. The percent compliance with the GDA increased from 15% in phase I to 38% in phase II. The average duration of therapy in phase I was 4.77 days (ranging from 1-16) vs 2.69 days in phase II (ranging from 1-9). The percentage of patients continued on vancomycin beyond 48 hours decreased from 81% in phase I to 34% in phase II. A total of 18 pharmacist interventions were made in phase II with an acceptance rate of 100%. Discussion: Pharmacist intervention had an impact in increasing compliance to national guideline recommendations and decreasing the duration of empiric vancomycin therapy in patients with FN

Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets

In this note, we demonstrate a first-of-its-kind provable convergence of SGD to the global minima of appropriately regularized logistic empirical risk of depth $2$ nets -- for arbitrary data and with any number of gates with adequately smooth and bounded activations like sigmoid and tanh. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence of Frobenius norm regularized logistic loss functions on constant-sized neural nets which are "Villani functions" and thus be able to build on recent progress with analyzing SGD on such objectives.Comment: 18 Pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:2210.1145