Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization

We study stochastic gradient descent for solving conditional stochastic optimization problems, in which an objective to be minimized is given by a parametric nested expectation with an outer expectation taken with respect to one random variable and an inner conditional expectation with respect to the other random variable. The gradient of such a parametric nested expectation is again expressed as a nested expectation, which makes it hard for the standard nested Monte Carlo estimator to be unbiased. In this paper, we show under some conditions that a multilevel Monte Carlo gradient estimator is unbiased and has finite variance and finite expected computational cost, so that the standard theory from stochastic optimization for a parametric (non-nested) expectation directly applies. We also discuss a special case for which yet another unbiased gradient estimator with finite variance and cost can be constructed.Comment: minor revision, 24 pages, 4 figure

Well-Differentiated Liposarcoma, an Atypical Lipomatous Tumor, of the Mesentery: A Case Report and Review of the Literature

Mesenteric liposarcoma is a rare neoplasm. Here, we report the case of a 73-year-old Japanese man with a well-differentiated (WD) liposarcoma of the mesentery. Due to rapid growth of the abdominal mass and abdominal insufficiency, a tumorectomy was performed. The excised tumor was 12.4 × 9.6 cm in size and weighed 548 g. Cut sections showed a lobulated yellow and/or grayish-colored appearance. The histological features were predominantly those of the sclerotic and lipoma-like variants of WD liposarcoma. The cytoplasm of most spindle cells was diffusely immunoreactive for CD34, while fat cells were positive for S-100 protein. Some spindle cell nuclei were positive for CDK4, and a few were positive for MDM2. The average Ki-67 proliferation index in tumor cells was 10%, and androgen receptor expression was detected in tumor cell nuclei. The present case and 11 cases identified from a literature search were reviewed. The WD mesenteric liposarcomas developed in patients in the fourth to seventh decades of life (mean age 57.9 years). The patients consisted of 7 men and 5 women. All tumors were larger than 10 cm in diameter at the time of surgery. Complete resection might be the only curative therapy for WD liposarcomas of the mesentery, but long-term follow-up is needed because of the possibility of a local recurrence of the tumor

Yキャク フンゴウブ キョウサク オ ガッペイ シタ チョウ コウレイシャ イ ゼンテキ ノ イチレイ

An 88-year-old male patient was referred to our hospital after being diagnosed with gastric cancer during extensive investigations for anemia. The patient underwent total gastrectomy followed by Roux-en Y reconstruction. An esophagus jejunum anastomosis was performed using a Curbed-Shaft Detachable Head Circular Stapler （25 mm）（CDH25）. An anastomosis of the jejunum was performed approximately 40 cm distal to the esophagus jejunum anastomosis using a Straight Shaft Detachable Head Circular Stapler（21 mm）（SDH21）. A seromuscular suture was applied to each anastomotic site. The patient started oral intake on Day 8 after the operation, and was transferred to the urology department on Day 22 for the treatment of renal cancer. Approximately 2 weeks after the transfer, the patient developed fever above 38℃. CT revealed marked enlargement of the duodenum, suggesting stenosis at the Y anastomotic site. The stenosis was successfully treated by endoscopic balloon dilatation without performing. Here, we report a case with favorable outcome

Surgical Control of a Primary Hepatic Carcinoid Tumor: A Case Report

We report a primary hepatic carcinoid tumor occurring in a 47-year-old man. The patient consulted our hospital complaining of epigastralgia. Abdominal ultrasonography, computed tomography scanning, and magnetic resonance imaging showed a large mass in the right lobe of the liver. FDG-PET revealed 18F-FDG uptake by the right hepatic lobe. The tumor was a solid mass with cystic components, approximately 15 cm in diameter. We conducted an extended right lobectomy of the liver. The resected specimen was a solid tumor with cystic components and hemorrhagic lesion. Microscopic findings showed that the tumor cells had round nuclei and formed trabecular patterns. Immunohistologically, tumor cells were stained positive for chromogranin A, neuron specific enolase, CD56, and S-100. Careful examinations before and after the operation revealed no other possible origin of the tumor. Based on these findings, the tumor was diagnosed as a primary hepatic carcinoid. This is a report of a rare case of a primary hepatic carcinoid tumor with a discussion of several other relevant reports